John Calvin, in his argument against the role of the Church in the canonical process, does discuss the role of the Holy Spirit. However, he seems to indicate that the Holy Spirit works in the individual, but not in and through the Church.
A most pernicious error has very generally prevailed—viz. that Scripture is of importance only in so far as conceded to it by the suffrage of the Church; as if the eternal and inviolable truth of God could depend on the will of men. With great insult to the Holy Spirit, it is asked, who can assure us that the Scriptures proceeded from God; who guarantee that they have come down safe and unimpaired to our times; who persuade us that this book is to be received with reverence, and that one expunged from the list, did not the Church regulate all these things with certainty? On the determination of the Church, therefore, it is said, depend both the reverence which is due to Scripture, and the books which are to be admitted into the canon. (Calvin, The Institutes of the Christian Religion 2005, 74-75)
Calvin then argues that since the apostles and prophets existed prior to the Church, that the inspiration of the Scriptures is intrinsic apart from the Church.
These ravings are admirably refuted by a single expression of an apostle. Paul testifies that the Church is “built on the foundation of the apostles and prophets,” (Eph. 2:20). If the doctrine of the apostles and prophets is the foundation of the Church, the former must have had its certainty before the latter began to exist. Nor is there any room for the cavil, that though the Church derives her first beginning from thence, it still remains doubtful what writings are to be attributed to the apostles and prophets, until her Judgment is interposed. For if the Christian Church was founded at first on the writings of the prophets, and the preaching of the apostles, that doctrine, wheresoever it may be found, was certainly ascertained and sanctioned antecedently to the Church, since, but for this, the Church herself never could have existed. Nothings therefore can be more absurd than the fiction, that the power of judging Scripture is in the Church, and that on her nod its certainty depends. (Calvin, The Institutes of the Christian Religion 2005, 75)
John Calvin is correct that the inspiration of the Scriptures precedes its recognition by the Church. But if the Church’s determination of the canon is invalid, what does John Calvin offer in its place? Why, the Holy Spirit who enlightens the individual believer’s heart.
Let it therefore be held as fixed, that those who are inwardly taught by the Holy Spirit acquiesce implicitly in Scripture; that Scripture, carrying its own evidence along with it, deigns not to submit to proofs and arguments, but owes the full conviction with which we ought to receive it to the testimony of the Spirit. Enlightened by him, we no longer believe, either on our own Judgment or that of others, that the Scriptures are from God; but, in a way superior to human Judgment, feel perfectly assured—as much so as if we beheld the divine image visibly impressed on it—that it came to us, by the instrumentality of men, from the very mouth of God. We ask not for proofs or probabilities on which to rest our Judgment, but we subject our intellect and Judgment to it as too transcendent for us to estimate.
Such, then, is a conviction which asks not for reasons; such, a knowledge which accords with the highest reason, namely knowledge in which the mind rests more firmly and securely than in any reasons; such in fine, the conviction which revelation from heaven alone can produce. I say nothing more than every believer experiences in himself, though my words fall far short of the reality. I do not dwell on this subject at present, because we will return to it again: only let us now understand that the only true faith is that which the Spirit of God seals on our hearts. (Calvin, The Institutes of the Christian Religion 2005, 78-79)
It is curious that John Calvin reason’s his way to a dismissal of human reason, but instead posits some ephemeral, mystical revelation of inspiration to the individual believer. Of course, John Calvin then modifies this by reference to the “children of the renovated Church” made up of the “elect only”, who “shall be taught of the Lord” (Isaiah 54:13). So Calvin’s argument isn’t so much against the Church bearing witness to the canon of Scripture, but to the Roman Catholic Church bearing said witness.
In essence, John Calvin’s predisposition against the Roman Catholic Church colors his view of canonicity. We can break down his argument like this: 1) The Holy Spirit works within His true church. 2) The Roman Catholics do not constitute a true Church. 3) Therefore, the Holy Spirit does not work within the Roman Catholic Church. Calvin makes another argument: 1) The Holy Spirit works upon the hearts of the elect. 2) The Roman Catholic Church contains none of the elect. 3) Therefore, the Holy Spirit does not work within the Roman Catholic Church. And finally, with regard to the canon of Scripture: 1) The Holy Spirit works to reveal the canon of Scripture to His Church. 2) The Roman Catholic Church is not a true Church. 3) Therefore, the Roman Catholic canon of Scripture was not revealed by the Holy Spirit.
Calvin, John. The Institutes of the Christian Religion. Translated by Henry Beveridge. Grand Rapids: Christian Classics Ethereal Library, 2005.
 I may not have constructed these syllogisms correctly, but you get the point.