The riposte from Roman Catholic and Orthodox apologists, maybe even some Anglican, is that the Bible’s canonicity is given to it in a causal way by the Church’s recognition. But the historic 
Protestant response is that as Calvin emphasizes, it is autopistis; or in other words, it is self-authenticating. Karl Barth, often chastised for being an enemy of Scripture, is actually one of Scripture’s most ardent warriors; particularly in regard to its Protestant iteration. Barth appeals to historic Protestant thinkers in his own articulation of a doctrine of Scripture. More pointedly, Barth appeals to John Gerhard and Johannes Wollebius as they counter Roman Catholics like, Sylvester Prierias and John Eck when they grant infallibility to the Pope and Roman Catholic Church over against Holy Scripture. Clearly this is an issue of authority; who has it? Does Scripture have it at a formal level, or does the Holy Roman See (this not to mention Orthodox conceptions of episcopacy vis-à-vis Scripture)?
Barth quotes Gerhard:
There is not a double authority of Scripture, but a single authority, and that is divine: it does not depend on the authority of the church, but on God alone. The authority of Scripture as far as we are concerned is nothing other than the manifestation and the knowledge of that single divine and supreme authority, which is internal and intrinsic to Scripture. Therefore, the church does not confer a new authority, as far as we are concerned, upon Scripture, but rather by its own testimony it leads us to the acknowledgement of that truth. We admit that of sacred Scripture the church is (1) the witness; (2) the guardian; (3) protector (4) the herald; (5) the interpreter. But let us not conclude from this that the authority of Scripture, whether in itself or in relation to us, depends on the church.[1]
And Wollebius:
The testimony of the church is prior in time, but the testimony of the Holy Spirit is prior in nature and causality. We believe that the testimony is of the church, not on account of the church. It is to be ascribed to the Holy Spirit on his own account. The testimony of the church demonstrates the fact ‘that’, but the testimony of the Holy Spirit demonstrates the ‘because’. The church advises, the Holy Spirit convinces. The testimony of the church provides opinion, but the testimony of the Holy Spirit provides knowledge and firm trustworthiness.[2]
Some Protestants, typically biblical studies folks it seems, want to signal the end of the Protestant Reformation. That in light of the New Paul Perspective[s] the soteriological reasons for the Protestant Reformation are no longer relevant; that Luther et al. were misguided. But these folks are moving too quickly. The Protestant Reformation, indeed had much to do with soteriology, but behind that there was a deeper issue of authority. This issue is ultimately an ecclesiological issue, and one that remains. What Gehard and Wollebius wrote in the 17th century is just as pertinent now as it was then; none of these issues have been resolved. Protestants still affirm the ‘Scripture Principle’ or sola Scriptura, whereas Roman Catholics (and Orthodox) do not. Indeed, at this level, the traditions are as far apart as ever. And this does have serious soteriological implications. If Rome or Constantinople have ‘the keys,’ then they decide who is genuinely ‘saved’ and who isn’t. There might be room for ecumenicism, such as we find with Thomas Torrance and the Orthodox focused on the doctrine of Trinity. But if we were to look to a doctrine of Scripture as the basis for ecumenical convergence between the traditions, it is not there.
[1] John Gerhard, Loci theolo., 1610 f., LI c. 3, 39, cited by Karl Barth, CD I/2 §19, 19.
[2] Johannes Wollebius, Comp. Christ. Theol., 1620, Praecogn. 9, cited by Karl Barth, CD I/2 §19, 19.