Quaeritur: It would seem like the Second Vatican Council is infallible, since the Catechism of the Catholic Church (CCC 891) states:
The Roman Pontiff, head of the college of bishops, enjoys this infallibility in virtue of his office, when, as supreme pastor and teacher of all the faithful - who confirms his brethren in the faith he proclaims by a definitive act a doctrine pertaining to faith or morals. The infallibility promised to the Church is also present in the body of bishops when, together with Peter's successor, they exercise the supreme Magisterium, above all in an Ecumenical Council.
Respondeo: Fr. Romanus and I answered this question a few years back in six posts (and then some), but I will take the liberty to answer it again, this time more succinctly. The text from the CCC means that a Council can be infallible, not that everything in every council is infallible. Indeed, the canons and decrees of all previous councils are infallible, but not absolutely everything that is said in every council. Stated simply, the reason is this: one of the conditions for a magisterial statement to be infallible is that it is binding (for when the Church binds us to believe or assent, she guarantees with her charism of infallibility that she is right). And for her to bind us to believe, or assent to, something, she must explicitly propose her teaching as binding. This is the traditional doctrine and practice of the Church, specifically the practice of all twenty one ecumenical councils, a doctrine that even Vatican II (in the well-known nota praeva to Lumen gentium) itself reiterates:
In view of the conciliar practice and the pastoral purpose of the present Council, this sacred Synod defines matters of faith or morals as binding on the Church only when the Synod itself openly declares so.
As a matter of fact, nowhere in the council documents does the Synod openly declare that such and such a doctrine is being defined.
That alone should be sufficient internal evidence, but, in fact, there plenty of external evidence in authoritative magisterial, papal and episcopal texts that clearly indicate that the Council was not infallible, and did not wish to define any doctrines. For example, Pope Paul VI, at the close of the council said:
Today we are concluding the Second Vatican Council.... But one thing must be noted here, namely, that the teaching authority of the Church, even though not wishing to issue extraordinary dogmatic pronouncements, has made thoroughly known its authoritative teaching on a number of questions which today weigh upon man’s conscience and activity, descending, so to speak, into a dialogue with him, but ever preserving its own authority and force; it has spoken with the accommodating friendly voice of pastoral charity; its desire has been to be heard and understood by everyone; it has not merely concentrated on intellectual understanding but has also sought to express itself in simple, up-to-date, conversational style, derived from actual experience and a cordial approach which make it more vital, attractive and persuasive; it has spoken to modern man as he is. (Address at the last general meeting of the Second Vatican Council, December 7, 1965; AAS 58)
After the Council, Pope Paul also said:
There are those who ask what authority, what theological qualification, the Council intended to give to its teachings, knowing that it avoided issuing solemn dogmatic definitions backed by the Church’s infallible teaching authority. The answer is known by those who remember the conciliar declaration of March 6, 1964, repeated on November 16, 1964. In view of the pastoral nature of the Council, it avoided proclaiming in an extraordinary manner any dogmas carrying the mark of infallibility. (General Audience, December 1, 1966; L’Osservatore Romano, January 21,1966)
The 'declaration' he is speaking of is the text of the nota praevia to Lumen gentium quoted above.
I think this all establishes pretty clearly that the Council is not, nor was ever intented to be, dogmatically binding or infallible, since nothing in it was defined as de fide. Rather, the kind of submission due to it is that of the next level down, namely, the "religious submission of mind and will" of which the Council itself speaks in Lumen Gentium 25 and which is reiterated in the 1983 Code of Canon Law:
Can. 752. While the assent of faith is not required, a religious submission of intellect and will is to be given to any doctrine which either the Supreme Pontiff or the College of Bishops, exercising their authentic magisterium, declare upon a matter of faith or morals, even though they do not intend to proclaim that doctrine by definitive act. Christ’s faithful are therefore to ensure that they avoid whatever does not accord with that doctrine.