No, Calvin was not Nestorian. This can be concluded on the strength of the following evidence:
His defense of the Chalcedonian Definition and rejection of Nestorius
Related to the above, reformed theology's rejection of icons of Christ on the basis of the unity of the natures
His implicit acceptance of the reality described in the term theotokos, and his rejection of the use of the term for different reasons than Nestorius
The appearance of the so-called extra calvinisticum in firmly orthodox (Ephesian, Alexandrine, Chalcedonian) church fathers
The appearance, in substance, of finitum non capax infiniti in Aquinas
The fact that he agrees with Nestorius' second anathema no more than Cyril did
The fact that he disagrees with Cyril's eleventh anathema for entirely different reasons than Nestorius
Reformed theology's acceptance of the unity of Christ's natures in death
Calvin on the unity of Christ
In Calvin's commentary on John, he gives a classic summary of the Chalcedonian Definition, and in the process condemns Nestorius.
The plain meaning therefore is, that the Logos begotten by God before all ages, and who always dwelt with the Father, was made man. On this article there are two things chiefly to be observed. The first is, that two natures were so united in one Person in Christ, that one and the same Christ is true God and true man. The second is, that the unity of person does not hinder the two natures from remaining distinct, so that his Divinity retains all that is peculiar to itself, and his humanity holds separately whatever belongs to it. And, therefore as Satan has made a variety of foolish attempts to overturn sound doctrine by heretics, he has always brought forward one or another of these two errors; either that he was the Son of God and the Son of man in so confused a manner, that neither his Divinity remained entire, nor did he wear the true nature of man; or that he was clothed with flesh, so as to be as it were double, and to have two separate persons. Thus Nestorius expressly acknowledged both natures, but imagined two Christs, one who was God, and another who was man. Eutyches, on the other hand, while he acknowledged that the one Christ is the Son of God and the Son of man, left him neither of the two natures, but imagined that they were mingled together. And in the present day, Servetus and the Anabaptists invent a Christ who is confusedly compounded of two natures, as if he were a Divine man. In words, indeed, he acknowledges that Christ is God; but if you admit his raving imaginations, the Divinity is at one time changed into human nature, and at another time, the nature of man is swallowed up by the Divinity.
The Evangelist says what is well adapted to refute both of these blasphemies. When he tells us that the Logos was made flesh, we clearly infer from this the unity of his Person; for it is impossible that he who is now a man could be any other than he who was always the true God, since it is said that God was made man. On the other hand, since he distinctly gives to the man Christ the name of the Logos, it follows that Christ, when he became man, did not cease to be what he formerly was, and that no change took place in that eternal essence of God which was clothed with flesh. In short, the Son of God began to be man in such a manner that he still continues to be that eternal Logos who had no beginning of time.
(Commentary on John 1:14)
Elsewhere he affirms that Christ the man was God, and again condemns Nestorius.
Because there be distinct natures in Christ, the Scripture cloth sometimes recite that apart by itself which is proper to either. But when it setteth God before us made manifest in the flesh, it doth not separate the human nature from the Godhead. Notwithstanding, because again two natures are so united in Christ, that they make one person, that is improperly translated sometimes unto the one, which doth truly and in deed belong to the other, as in this place Paul doth attribute blood to God; because the man Jesus Christ, who shed his blood for us, was also God. This manner of speaking is caned, of the old writers, communicatio idiomatum, because the property of the one nature is applied to the other. And I said that by this means is manifestly expressed one person of Christ, lest we imagine him to be double, which Nestorius did in times past attempt.
(Commentary on Acts 20:28)
Reformed iconoclasm and Nestorianism
Historically, iconoclasts accuse iconophiles of Nestorianism. This can be seen in the Council of Hieria, which condemned icons before being declared a "Robber Council" at the Second Council of Nicea. It can also be seen in today's reformed theologians. They read that iconophiles defend icons of Christ as depicting his human nature and not his divine nature, and they object, "But the natures are inseparable! Jesus is God! To divide the natures is Nestorian."
Calvin and the theotokos
The Nestorian controversy centered largely around the use of theotokos to describe Mary. Calvin never used the term, but it seems agreeable to what he says here:
The blessedness brought to us by Christ cannot be the subject of our praise, without reminding us, at the same time, of the distinguished honor which God was pleased to bestow on Mary, in making her the mother of his Only Begotten Son. ... She [Elizabeth] calls Mary the mother of her Lord. This denotes a unity of person in the two natures of Christ; as if she had said, that he who was begotten a mortal man in the womb of Mary is, at the same time, the eternal God. For we must bear in mind, that she does not speak like an ordinary woman at her own suggestion, but merely utters what was dictated by the Holy Spirit.
(Commentary on Luke 1:42-43)
To paraphrase: God made Mary the mother of his Son, that is, the Son of God, the second person of the trinity. Jesus, who is in her womb, is "the eternal God."
That's a far cry from Nestorius, who rejected the term "theotokos" as false on the basis that Jesus was not actually God but "the temple of the Godhead," as the Catholic Encyclopedia explains:
We have no difficulty in defining the doctrine of Nestorius so far as words are concerned: Mary did not bring forth the Godhead as such (true) nor the Word of God (false), but the organ, the temple of the Godhead. The man Jesus Christ is this temple, "the animated purple of the King", as he expresses it in a passage of sustained eloquence. The Incarnate God did not suffer nor die, but raised up from the dead him in whom He was incarnate.
Calvin's protege Heinrich Bullinger explicitly used the term. To the extent that their theology was in harmony (which is a great extent, considering Bullinger authored the reformed Second Helvetic Confession):
The Virgin Mary ... completely sanctified by the grace and blood of her only Son and abundantly endowed by the gift of the Holy Spirit and preferred to all ... now lives happily with Christ in heaven and is called and remains ever-Virgin and Mother of God.
[I haven't yet found a primary source for this. I found this on a website that quotes a secondary source. I have that source on order from the library, and will amend this answer once I've investigated the quote.]
Calvin did say that he didn't think the term theotokos was beneficial, but for much different reasons than Nestorius:
I cannot think such language either right, or becoming, or suitable. ... To call the Virgin Mary the mother of God can only serve to confirm the ignorant in their superstitions.
Epistle CCC to the French church in London
Notice, he thought it would "confirm the ignorant in their superstitions," i.e. veneration of Mary as mediatrix. Again, a far cry from, "Jesus was the temple of the Godhead, not actually God."
Calvin would (I'm speaking hypothetically, but with confidence nonetheless) similarly reject calling David the "Father of God" even though he truly is Jesus' forefather, or calling anyone an "ancestor of God." Jesus apprenticed under Joseph, making Joseph "master of God" or "teacher of God." Each of these expressions could possibly have an orthodox use, by stirring up awe and wonder about God becoming a man, but they could just as likely confuse people unto superstition. This seems to be what Calvin was telling the London church to guard against.
Notice that he rejects it because it "can only serve to confirm the ignorant in their superstitions," not because the man/baby Jesus wasn't properly called God.
Extra-Calvinistic precedent for the extra calvinisticum
Try to find any inconsistency between these words of the church fathers (including Cyril himself) and the definition of the extra calvinisticum provided above:
The Word was not hedged in by His body, nor did His presence in the body prevent His being present elsewhere as well. When He moved His body He did not cease also to direct the universe by His Mind and might. No. The marvelous truth is that being the Word, so far from being Himself contained by anything, He actually contained all things Himself. In creation He is present everywhere, yet is distinct in being from it; ordering, directing, giving life to all, containing all, yet is He Himself the Uncontained, existing solely in His Father…. His body was for Him not a limitation, but an instrument, so that He was both in it and in all things, and outside all things, resting in the Father alone. At one and the same time – this is the wonder – as Man He was living a human life, and as Word He was sustaining the life of the universe, and as Son He was in constant union with the Father.
(Athanasius, On the Incarnation 17)
We confess that the Only begotten Word of God, begotten of the same substance of the Father, True God from True God, Light from Light, through Whom all things were made, the things in heaven and the things in the earth, coming down for our salvation, making himself of no reputation, was incarnate and made man; that is, taking flesh of the holy Virgin, and having made it his own from the womb, he subjected himself to birth for us, and came forth man from a woman, without casting off that which he was; but although he assumed flesh and blood, he remained what he was, God in essence and in truth. Neither do we say that his flesh was changed into the nature of divinity, nor that the ineffable nature of the Word of God was laid aside for the nature of flesh; for he is unchanged and absolutely unchangeable, being the same always, according to the Scriptures. For although visible and a child in swaddling clothes, and even in the bosom of his Virgin Mother, he filled all creation as God, and was a fellow-ruler with him who begat him, for the Godhead is without quantity and dimension, and cannot have limits.
(Cyril of Alexandria, Third Letter to Nestorius)
The Infinite God, remaining changeless, assumed flesh and fought with death, freeing us from all suffering by his own suffering.
(Basil the Great, On the Holy Spirit, 8.18)
The Christian doctrine does not hold that the Godhead was so blended with the human nature in which He was born of the virgin that He either relinquished or lost the administration of the universe, or transferred it to that body as a small and limited material substance.
(Augustine, Letter 137 to Volusianus)
It is not to be believed that this form [Christ's body] is everywhere diffused. For we have to be careful lest we so construct the divinity of the man that we give up the truth of the body. For it does not follow that what is in God is everywhere as God is everywhere.
(Augustine, Letter 187 to Dardanus, quoted in Edward David Willis, Calvin's Catholic Christology, 29. If I'm reading it correctly, Willis says that an older edition of Calvin's Institutes quoted this passage. I have not been able to locate anything else about it online.)
Aquinas and the non capax
Similarly, how can one distinguish these words of Aquinas from the dictum finitum non capax infiniti?
As is plain from 2, 1,6, the union of the two natures in the Person of Christ took place in such a way that the properties of both natures remained unconfused, i.e. "the uncreated remained uncreated, and the created remained within the limits of the creature," as Damascene says (De Fide Orth. iii, 3,4). Now it is impossible for any creature to comprehend the Divine Essence, as was shown in I, 12, 1,4,7, seeing that the infinite is not comprehended by the finite. And hence it must be said that the soul of Christ nowise comprehends the Divine Essence.
(Aquinas, Summa Theologica, 3,Q10,A1)
Nestorius' second anathema
It does appear that Calvin would agree with Nestorius on the second anathema:
If any one asserts that, at the union of the Logos with the flesh, the divine Essence moved from one place to another; or says that the flesh is capable of receiving the divine nature, and that it has been partially united with the flesh; or ascribes to the flesh, by reason of its reception of God, an extension to the infinite and boundless, and says that God and man are one and the same in nature; let him be anathema.
However, this is not the teaching of Nestorius that Cyril condemned. In fact, if it were so, Cyril and the Council of Ephesus would be affirming that, "God and man are one and the same in nature." Plainly, that is not the case. Nestorius, however, in this anathema, was responding to this one from Cyril:
If anyone shall not confess that the Word of God the Father is united hypostatically to flesh, and that with that flesh of his own, he is one only Christ both God and man at the same time: let him be anathema.
This, too, is doctrine that Calvin would affirm. But the difference is that Nestorius would not.
Cyril's eleventh anathema
Cyril's defense of eucharistic adoration is predicated on a belief in the Real Presence, which Nestorius evidently affirmed, judging by this entry in the Catholic Encyclopedia: "the Nestorians ... preserved their faith in the Eucharist as unwaveringly as the Greeks."
What Cyril appears to be saying is, "If Christ is present as a man in the host, how could he not also be present as God?" which Calvin would not disagree with (he says in his Institutes that "the whole Christ is offered in the supper.") Calvin simply disagrees with the premise. Yes, he seems to disagree with Cyril here, but since that disagreement isn't shared with Nestorius, it doesn't make sense to refer to it as Nestorianism.
The unity of Christ's natures in death
On another answer you linked to an R.C. Sproul article where he said that God cannot be said to have died on the cross. But note that Sproul gave a qualifier: "If we say that God died on the cross, and if by that we mean that the divine nature perished, we have stepped over the edge into serious heresy." If we mean the divine nature perished. That, I think, is agreeable to all Christians. Kevin DeYoung similarly says:
You can say, for example, "God died" if you mean that God as a man in the person of Jesus Christ died. But you can‘t say "God died" if you mean God as God died.
Note also that Sproul himself affirms that Mary is the theotokos.
And the Belgic Confession, a confessional statement of Calvinism written by Calvin's student Guido de Bres, says in article 19:
We believe that by being thus conceived the person of the Son has been inseparably united and joined together with human nature, in such a way that there are not two Sons of God, nor two persons, but two natures united in a single person, with each nature retaining its own distinct properties.
Thus his divine nature has always remained uncreated, without beginning of days or end of life, filling heaven and earth.
Christ’s human nature has not lost its properties but continues to have those of a creature—it has a beginning of days; it is of a finite nature and retains all that belongs to a real body. And even though he, by his resurrection, gave it immortality, that nonetheless did not change the reality of his human nature; for our salvation and resurrection
depend also on the reality of his body.
But these two natures are so united together in one person that they are not even separated by his death.
So then, what he committed to his Father when he died was a real human spirit which left his body. But meanwhile his divine nature remained united with his human nature even when he was lying in the grave; and his deity never ceased to be in him, just as it was in him when he was a little child, though for a while it did not so reveal itself.
These are the reasons why we confess him to be true God and truly human—true God in order to conquer death by his power, and truly human that he might die for us in the weakness of his flesh.
Who died on the cross? Jesus Christ, who was both God and man.
Additional Reading and Footnotes
