Can a Pope be excommunicated?
According to Scripture, a heretic definetly can be excommunicated
A man that is an heretick after the first and second admonition reject [Titus 3:10]
According to Canon, a Pope who is heretic, theoretically can be excommunicated. (Can. 1371)
Thomas Aquinas in his Commentary on St. Paul’s Epistle to Titus said: “Such a person should be warned, and if he does not desist, he should be avoided. And he [the Apostle] says, after the first and second admonition, for that is the way the Church proceeds in excommunicating”
But the Church must do this admonitions, not the mere believers
On the part of the Church, however, there is mercy which looks to the conversion of the wanderer, wherefore she condemns not at once, but ‘after the first and second admonition,’ as the Apostle directs: after that, if he is yet stubborn, the Church no longer hoping for his conversion, looks to the salvation of others, by excommunicating him and separating him from the Church, and furthermore delivers him to the secular tribunal to be exterminated thereby from the world by death.(ST, II-II, q. 11, a. 3, sed contra., St. Thomas Aquinas)
Also William of Ockham wrote a treatise called Dialogus inter magístrum et disculum de imperatórum et pontíficum potestáte (Dialogue between a teacher and his disciple on the power of the Emperor and the Pontiff), versed on heresy and the possibility that a Pope could fall into heresy, from which it is inferred that the aid of infallibility is only by the papal office and in favor of the defense of the Faith. A situation in which he is ipso facto and latae sententiae deposed from his authority and his subjects have the obligation to resist him and fight against him.
About how it should be done
The Pope is the head of the church and has no bishops above him (Can. 331), so presenting these warnings are a mere act of charity.
A subject is not competent to administer to his prelate the correction which is an act of justice through the coercive nature of punishment: but the fraternal correction which is an act of charity is within the competency of everyone in respect of any person towards whom he is bound by charity, provided there be something in that person which requires correction. (ST, II-II, q. 33, a. 4., Thomas Aquinas)
For a lay believer or a lesser bishop or parish priest, it is not an obligation to present these precautions prior to his excommunication as a Catholic for heresy.
Neither is it always demanded in the external forum that there be a warning and a reprimand as described above for somebody to be punished as heretical and pertinacious, and such a requirement is by no means always admitted in practice by the Holy Office. (De Lugo, disp. XX, sect. IV, n. l57-158, cited in “Essay on Heresy,” by Arnaldo da Silveira).
[H]e is not excommunicated on account of heresy, but should be excommunicated by being deposed. Therefore, the apostle’s command concerning the double admonition, which need not be observed [to the letter] in the case of others, who are inferiors, on account of the addition of excommunication latae sententiae, which the Church imposes on heretics, should be observed to the letter with him. (De Comparatione Auctoritatis Papae et Concilii, p. 103. Cajetan)
However, it is possible to submit these admonitions to a Pope who is violating the Canon Law (Can. 1369 A person who in a public show or speech, in published writing, or in other uses of the instruments of social communication utters blasphemy, gravely injures good morals, expresses insults, or excites hatred or contempt against religion or the Church is to be punished with a just penalty.), and above all the Holy Scripture.
Is it not true that, confronted with such a danger to the faith [a Pope teaching heresy], any subject can, by fraternal correction, warn their superior, resist him to his face, refute him and, if necessary, summon him and press him to repent? The Cardinals, who are his counselors, can do this; or the Roman Clergy, or the Roman Synod, if, being met, they judge this opportune. For any person, even a private person, the words of Saint Paul to Titus hold: ‘Avoid the heretic, after a first and second correction, knowing that such a man is perverted and sins, since he is condemned by his own judgment’ (Tit. 3, 10-11). For the person, who, admonished once or twice, does not repent, but continues pertinacious in an opinion contrary to a manifest or defined dogma – not being able, on account of this public pertinacity to be excused, by any means, of heresy properly so called, which requires pertinacity – this person declares himself openly a heretic. He reveals that by his own will he has turned away from the Catholic Faith and the Church, in such a way that now no declaration or sentence of anyone whatsoever is necessary to cut him from the body of the Church. Therefore the Pontiff who after such a solemn and public warning by the Cardinals, by the Roman Clergy or even by the Synod, would remain himself hardened in heresy and openly turn himself away from the Church, would have to be avoided, according to the precept of Saint Paul. So that he might not cause damage to the rest, he would have to have his heresy and contumacy publicly proclaimed, so that all might be able to be equally on guard in relation to him. Thus, the sentence which he had pronounced against himself would be made known to all the Church, making clear that by his own will he had turned away and separated himself from the body of the Church, and that in a certain way he had abdicated the Pontificate (De Potestate Ecclesiastica, (Monasterii Westphalorum, Deiters, 1847) ch. 6, sec. 2, pp. 124-125)
Thus, before the Pope could be considered a public heretic, he would have to be issued a first and a second warning by the Cardinals or other official Church authority, such as a Roman Synod.
I affirm: if he were a heretic and incorrigible the Pope would cease to be Pope just when a sentence was passed against him for his crime, by the legitimate jurisdiction of the Church. This is the common opinion among the doctors (De Fide, disp. X, sect. VI, nn. 3-10, p. 316; Suarez)
By what power should a deposition happen with regard to the pope? The entire question hinges on two points, namely one, a declarative sentence, by which it is declared (. . .) that the pope has committed the crime (. . . ) and two, the deposition itself, which must be done after the declarative judgment of the crime. (. . .) The Church is able to declare the crime of a Pontiff and, according to divine law, propose him to the faithful as a heretic that must be avoided. (. . . ) the deposition of the pope with respect to the declaration of the crime in no way pertains to the cardinals but to a general council. (Cursus Theologici II-II, John of St. Thomas, De Auctoritate Summi Pontificis, Disp. II, Art. III, De Depositione)
To carry out any type of sentence, it is necessary to form a council, which can be carried out without the presence of the Pope.
A perfect council according to the present state of the Church [i.e., an imperfect council] can be summoned without the pope and against his will, if, although asked, he himself does not wish to summon it; but it does not have the authority to regulate the universal Church, but only to provide for the issue then at stake. Although human cases vary in infinite ways … there are only two cases that have occurred or can ever occur, in which, I declare, such a council should be summoned. The first is when the pope must be deposed on account of heresy; for then, if he refused, although asked, the cardinals, the emperor, or the prelates can cause a council to be assembled, which will not have for its scope the care of the universal Church, but only the power to depose the Pope. (De Comparatione Auctoritatis Papae et Concilii, p. 70, Cajetan)
So, this require the Church to judge and declare the Pope guilty of the crime of heresy (Can. 1364) before Christ would remove him from office in latae sententiae (automatically).
It is necessary that, just as the Church designates the man and proposes him to the faithful as being elected Pope, so too is it necessary that the Church declares him a heretic and proposes him as one to be avoided. Hence, we see from the practice of the Church that this is how it has been done; for, in the case of the deposition of a Pope, his cause was handled in a general Council before he was considered not to be Pope, as we have related above. Therefore, it is not because the Pope is a heretic, even publicly, that he will ipso facto cease to be Pope, before the declaration of the Church and before she proclaims him as ‘to be avoided’ by the faithful. (John of St. Thomas, Cursus Theologici II-II, On the Authority of the Supreme Pontiff, Disp. 2, Art. 3.)
Even if he repents of his sins after the council has been held, he could not return to occupy the position, because as 1 Timothy 3:2 states "A bishop then must be blameless..."
Can this Pope be excommunicated?
Sadly, probably no.
Although it would not seem necessary for even the majority, much less all, the cardinals and bishops to declare and prove the incorrigibility of the Pope regarding the crime of his heresies. [Because a legitimate imperfect council unites the faithful without the approval of the Pope, it follows that it would not require the approval or participation of all bishops, not even most of them.] The new cardinals are allies of Bergoglio (sources: 1, 2) and would not condemn him.
Nevertheless, he has been already condemned of heresy by many cardinals, bishops, priests, theologians and lay believers (sources: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11), but the ears are deaf to all complaints, not only from the Pope, some bishops and priests, but from many believers as well.
Unfortunately, although he were excommunicated and deposed from office, Peter Kwasniewski comments on the future of the position are pessimistic: “The implausibility of this approach is demonstrated by, among other signs, the infinitesimal success that conservatives have had in reversing the disastrous “reforms,” trends, habits, and institutions established in the wake of and in the name of the last council, with papal approbation or toleration”.
But let us remember what Thomas Aquinas says about accepting this kind of acts: "It is praiseworthy to be patient under one's own wrongs, but the height of impiety to dissemble injuries done to God" (Aquinas Ethicus: The Moral Teaching of St. Thomas, Vol. 2). And also those words of St. Jerome: "A dog may bark in his master's defense, and am I to stand by silent when God's holy name is blasphemed? I would sooner die than forbear to speak".
We can only pray and prepare for what is to come, as G.K. Chesterton said in the conclusion to his book Heretics.
The great march of mental destruction will go on. Everything will be denied. Everything will become a creed. It is a reasonable position to deny the stones in the street; it will be a religious dogma to assert them. It is a rational thesis that we are all in a dream; it will be a mystical sanity to say that we are all awake. Fires will be kindled to testify that two and two make four. Swords will be drawn to prove that leaves are green in summer. We shall be left defending, not only the incredible virtues and sanities of human life, but something more incredible still, this huge impossible universe which stares us in the face. We shall fight for visible prodigies as if they were invisible. We shall look on the impossible grass and the skies with a strange courage. We shall be of those who have seen and yet have believed. THE END