The documents of Vatican II don’t limit biblical inerrancy to religious truths necessary for salvation or even to religious matters in general.
The Constitution on Divine Revelation (Dei Verbum), states, "Therefore, since everything asserted by the inspired authors or sacred writers must be held to be asserted by the Holy Spirit, it follows that the books of Scripture must be acknowledged as teaching solidly, faithfully, and without error that truth which God wanted put into the sacred writings for the sake of our salvation" (11).
Proponents of "limited inerrancy" claim this last clause is restrictive: Inerrancy extends only to things pertaining to our salvation. Whether or not this is the case (such a reading isn’t required by the Latin), the "limited inerrancy" position is still weak.
First, even granting (though not conceding) that Dei Verbum restricts inerrancy to matters of salvation, this isn’t the same as limiting it to religious or moral truths. Historical or scientific assertions made "for the sake of our salvation" would be inerrant too.
Second, the theological commission at the Council stated that the term salutaris ("for the sake of our salvation") doesn’t mean that only the salvific truths of the Bible are inspired or that the Bible as a whole isn’t the Word of God. (See A. Grillmeier’s "The Divine Inspiration and Interpretation of Sacred Scripture" in H. Vorgrimler, ed., Commentary on the Documents of Vatican II, vol. III, p. 213.)
If the whole of Scripture is inspired, and if what the biblical writer asserts the Holy Spirit asserts, then, unless error is to be attributed to the Holy Spirit or unless the biblical authors assert only religious truths (which isn’t the case--some make historical assertions, such as the historical existence of Jesus), inerrancy can’t be limited to religious truths.
Third, the language of Dei Verbum 11 is taken directly from previous conciliar and papal teaching on the subject. The footnotes to this section refer to Leo XIII’s Providentissimus Deus and Pius XII’s Divino Afflante Spiritu, documents which reject the idea that inerrancy is limited to religious matters. It seems unlikely the Council would be teaching a position contrary to these documents.
Although inerrancy isn’t limited to religious truths which pertain to salvation but may include non-religious assertions by the biblical authors, this doesn’t mean Scripture is an inspired textbook of science or history. Inerrancy extends to what the biblical writers intend to teach, not necessarily to what they assume or presuppose or what isn’t integral to what they assert. In order to distinguish these things, scholars must examine the kind of writing or literary genre the biblical writers employ.