Before we are exploring whether or not St. Thomas of Aquinas deny the Holy Spirit proceeds immediately from the Father alone, firstly it is important to establish that by the doctrine of the procession of the Holy Spirit from the Father and the Son no Catholics deny the Father to be the principaliter of the Holy Spirit. Any Catholics associated the Father together with the Son as the one principle of the Holy Spirit without denying that the Father alone is the principaliter1 while the Son is a principle2, a distinction developed by St. Augustine of Hippo.
There are two texts from Summa Theologia that we will be focusing primarily. The first one is from the first book on question 36 at article 2,
For when the Lord says, "No one knoweth the Son, but the Father," the idea of the Son knowing Himself is not excluded. So therefore when we say that the Holy Ghost proceeds from the Father, even though it be added that He proceeds from the Father alone, the Son would not thereby be at all excluded.
St. Thomas of Aquinas, Summa Theologia, I:36:ii(a1).
In here we find a positive affirmation from the Angelic Doctor whereby there will not be any theological difficulty "even though it be added that [the Holy Spirit] proceeds from the Father alone." He is especially respected by his opponents particularly because he honor the interlocutors by representing them with their best arguments. He did this by showing that no contradiction can be found by those insisting that the Holy Spirit in a sense is eternally proceeding from the Father alone and in another sense to be from the Son as later will be shown to be clear.
To put to rest the two extremities among his interlocutors who emphasized the involvement of the Son in the eternal procession of the Holy Spirit at the expense of the sole monarchy of the Father and against those who attributed the procession from the Son in the economy alone to combat diarchy, Aquinas maintain an important distinction. A distinction that was introduced by Augustine to keep the involvement of the Son in the eternal procession of the Holy Spirit in balance with the sole monarchy of the Father who alone is the principaliter of the Holy Spirit. The second important text that explicitly address this distinction is from the same question 36 at article 3,
So, if we consider in the Father and the Son the power whereby the spirate (virtus qua spirant) the Holy Spirit, there is no mean (medium), for this is one and the same power (virtus est una et eadem). But if we consider the persons themselves spirating (spirantes), then, as the Holy Spirit proceeds both (communiter procedat) from the Father and from the Son, the Holy Spirit proceeds from the Father immediately (immediate), as from him, and mediately (mediate), as from the Son; and thus he is said to proceed from the Father through the Son (per filium), which has the same meaning.
ibid, ST, I:36:iii(a1).
In this second text, he addressed two important points. The first is the unity of father/son as the one spirator of their Holy Spirit. This is an important point that shows the unity not shared by father/spirit or son/spirit. This unity between the Father and the Son distinguishes the two from their Holy Spirit.3 At the Council of Florence the language "as from one principle" is a reflection of this unitive spiration from the Father and the Son, whereby the Holy Spirit is their Holy Spirit.
The second important point is the crux of the debate whereby he distinguished the immediate procession from the mediate. Had he has in mind that the procession from the Son is identical and in the same sense as from the Father, he would be enforcing the diarchical filioque or the eternal procession immediately from both. The mediation of the Son is a reflection to the fact that the Son is spirating His Spirit as He is receiving Him from the Father, or in another way to see it is that the Father is begetting the Son in order to send His Spirit to rest in the Son. Hence, the spiration of the Holy Spirit is not an after thought but rather inherent in the generation of the Son that He is being begetting so that through the mediation of the Son in that generation the Holy Spirit also proceeding in Him.
To remove any possible misrepresentation or misinterpretation on the doctrine of filioque, St. John Paul II in 1995 commissioned Pontifical Council for Promoting Christian Unity to release an authoritative magisterial statement:
On the basis of Jn. 15:26, this Symbol confesses the Spirit "to ek tou Patros ekporeuomenon" ("who takes his origin from the Father"). The Father alone is the principle without principle (arche anarchos) of the two other persons of the Trinity, the sole source (peghe) of the Son and of the Holy Spirit. The Holy Spirit, therefore, takes his origin from the Father alone (ek monou tou Patros) in a principal, proper, and immediate manner.
The Greek Fathers and the whole Christian Orient speak, in this regard, of the "Father's Monarchy," and the Western tradition, following St. Augustine, also confesses that the Holy Spirit takes his origin from the Father principaliter, that is, as principle (De Trinitate XV, 25, 47, P.L. 42, 1094-1095). In this sense, therefore, the two traditions recognize that the "monarchy of the Father" implies that the Father is the sole Trinitarian Cause (Aitia) or Principle (Principium) of the Son and the Holy Spirit.
This origin of the Holy Spirit from the Father alone as Principle of the whole Trinity is called ekporeusis by Greek tradition, following the Cappadocian Fathers. St. Gregory of Nazianzus, the Theologian, in fact, characterizes the Spirit's relationship of origin from the Father by the proper term ekporeusis, distinguishing it from that of procession (to proienai) which the Spirit has in common with the Son. "The Spirit is truly the Spirit proceeding (proion) from the Father, not by filiation, for it is not by generation, but by ekporeusis" (Discourse 39. 12, Sources chretiennes 358, p. 175). Even if St. Cyril of Alexandria happens at times to apply the verb ekporeusthai to the Son's relationship of origin from the Father, he never uses it for the relationship of the Spirit to the Son (c.f. Commentary on St. John, X, 2, P.G. 74, 910D; Ep 55, P.G. 77, 316D, etc.). Even for St. Cyril, the term ekporeusis as distinct from the term "proceed" (proienai), can only characterize a relationship of origin to the principle without principle of the Trinity: the Father.
That is why the Orthodox Orient has always refused the formula to ek tou Patros kai tou Uiou ekporeuomenon [an unwisely proposed translation of "who proceeds from the Father and the Son"] and the Catholic Church has refused the addition kai tou Uiou [and the Son] to the formula ek to Patros ekporeumenon in the Greek text of the Nicene-Constantinopolitan Symbol, even in its liturgical use by Latins. ...
The doctrine of the Filioque must be understood and presented by the Catholic Church in such a way that it cannot appear to contradict the Monarchy of the Father nor the fact that he is the sole origin (arche, aitia) of the ekporeusis of the Spirit. ...
We are presenting here the authentic doctrinal meaning of the Filioque on the basis of the Trinitarian faith of the Symbol professed by the second Ecumenical Council at Constantinople. We are giving this authoritative interpretation, while being aware of how inadequate human language is to express the ineffable mystery of the Holy Trinity, one God, a mystery which is beyond our words and our thoughts.
PCPCU, The Father as the Source of the Whole Trinity: The Procession of the Holy Spirit in Greek and Latin Traditions (1995). Emphasis by the authors.
In 2003 the United States Conference of Catholic Bishops published an addendum to earlier 1995 Clarification on the Doctrine of Filioque,
No clear record exists of the process by which the word Filioque was inserted into the Creed of 381 in the Christian West before the sixth century. ... None of these writers, however, makes the Spirit’s mode of origin the object of special reflection; all are concerned, rather, to emphasize the equality of status of all three divine persons as God, and all acknowledge that the Father alone is the source of God’s eternal being. ...
Much of the difference between the early Latin and Greek traditions on this point is clearly due to the subtle difference of the Latin procedere from the Greek ekporeuesthai: as we have observed, the Spirit’s “coming forth” is designated in a more general sense by the Latin term, without the connotation of ultimate origin hinted at by the Greek. The Spirit’s “procession” from the Son, however, is conceived of in Latin theology as a somewhat different relationship from his “procession” from the Father, ...
[T]he Catholic Church, following a growing theological consensus, and in particular the statements made by Pope Paul VI, declare that the condemnation made at the Second Council of Lyons (1274) of those “who presume to deny that the Holy Spirit proceeds eternally from the Father and the Son” is no longer applicable.
USCCB, The Filioque: A Church Dividing Issue?: An Agreed Statement (2013).
Expanding upon its statement 'all acknowledge that the Father alone is the source of God’s eternal being,' it affirms that '[t]he Spirit’s “procession” from the Son, however, is conceived of in Latin theology as a somewhat different relationship from his “procession” from the Father.'
In regards to Medieval fathers, Dennis Ngien wrote:
Thus the filioque, far from dividing the Godhead into two separate sources, safeguards its fundamental unity. Even though the Spirit does not proceed from the Son in the same way as from the Father, it is vitally necessary that we name the Father and the Son together in affirming his procession; otherwise we might exclude the Spirit from the ontologically fundamental unity.
Dennis Ngien, Apologetic for Filioque in Medieval Theology, 31-2.
While there is only one spiration the procession is differentiated between immediate from the source and mediate from the channel, similar to how water propagate from the source through the river to the lake.4
1 The Father as the principle without principle in Latin is principium non de principio.
2 The Son is a principle from the principle, in Latin is principium de principio.
3 Because the Holy Spirit does not have the property to spirate, He has no paternal/filial namesake but rather called as a neuter common to the Father and the Son as their Holy Spirit.
4 Ss. John of Damascus and Anselm of Canterbury illustrated the eternal procession of the Holy Spirit from the Father through the Son with an analogy of water flow from the spring through the river to the lake.