There are really only two possible answers to this question. Either a Pope is able to fall into heresy, or he is not able. But some persons have claimed that a Pope can fall into heresy privately, but not in his public teaching, so we can divide the possibilities into three:
1. Can a Pope publicly teach heresy?
Suppose the position that a Pope can fall into heresy were true. A heretic cuts himself off from the Church by his sin. So the man would no longer be Pope, and would lose all this authority. Or so the theory says. But this proposition can be refuted by following it to its necessary conclusion.
Heresy can be committed either publicly, by proclaiming or even teaching heresy, or privately, hidden in the heart and mind. The idea that a Pope can possibly teach heresy, is contrary to the gift that Christ gave to the Church, the gift of the Magisterium. If we believe, from Sacred Tradition and Sacred Scripture that the Church has a teaching authority, guided by the Holy Spirit, an authority given to Her by Christ, resting on the Rock that is Peter and his successors, then we cannot believe that the Magisterium would ever teach heresy.
Now some persons claim that the Magisterium is unharmed by this possibility, since his teaching would be that of a person separated from the Church, and therefore not be of the Magisterium. But the harm to the faithful and to the Church as a whole would be immense. How would the faithful know if a teaching is heresy, if it is a teaching of the Pope? Would they know from past teachings of other Popes? But those Popes could be heretics also. Would they know from past teachings of Ecumenical Councils? But without a valid Pope as its head, no Council is a valid Council (cf. Universi Dominici Gregis, n. 34).
So then, would they know if a Pope is a heretic by their own interpretation of Sacred Tradition and Sacred Scripture? But interpretations vary; the faithful would not be able to agree, and would be shattered into pieces every time any Pope offers any teaching that is controversial. As each Pontificate and each century passes, there would accumulate so many teachings that, among fallen sinners, would seem arguable, that the Church would not survive.
The proposition cannot be true, because it would imply that Christ established the Church in such a manner that it could not stand, as if on sand, rather than on Rock. The indefectibility of the Church would be destroyed, so the claim is not true. A Pope cannot teach heresy.
2. Can a Pope privately believe heresy (but not teach it)?
Some persons claim that the Pope can fall into heresy privately, but cannot teach heresy publicly (or under the Magisterium). However, this idea has much the same effect on the Church as the previous proposition. Even if the Pope is prevented from teaching heresy, if he can fall into heresy in his heart and mind, he still cuts himself off from the Church by his sin. The result would be that all of this public teachings would not be an exercise of the Magisterium. Even if those public teachings are not heresy, they would not be teachings of the Magisterium, through the Holy Spirit; teachings of Christ.
And since, in this proposition, the heresy is hidden in the heart and mind, the faithful would have no way of knowing that the Pope had separated himself from the Church and was no longer teaching or acting with the authority of the head of the Church. Therefore, the faithful would have no way of knowing which Popes were heretics, which Councils were valid, which decisions under the temporal authority were valid, which papal teachings were of the Magisterium, throughout all of the history of the Church.
The result of this proposition is the same as in the previous proposition, the faithful would be scattered by every controversy about a teaching, by every controversy about a decision under the temporal authority; the Church would be utterly destroyed. Again, this cannot be the case, since the Church is indefectible.
Therefore, under both of the above propositions, the Magisterium would lose its authority.
Under the first proposition, no one would know which teachings of the Pope fell under the Magisterium, since they would not be able to agree as to whether one teaching or another is heresy. Each person is put in the position of judging the Pope’s teachings, to see if he is teaching heresy, but opinions among fallen sinners inevitably vary. So the Magisterium would lose its authority because no one would know teachings were of the Magisterium, and which were heresy.
Under the second proposition, no one would know which Popes were occult (hidden) heretics. If a man is not Pope, but rather a heretic, then he cannot exercise the Magisterium. So the result is the same, no one would know which teachings of the Pope were of the Magisterium, and which were the teachings of a heretic.
Since both propositions, in their end result, destroy the Church, neither can be true.
3. The only other possibility is that God positively prevents any Pope from falling into heresy, both in his public teachings and in his heart and mind. This gift to the Church protects the Magisterium and the indefectibility of the whole Church.
See my article Can a Pope Ever Be a Heretic? for a full explanation of why the Pope can never fall into heresy at all.
Ronald L. Conte Jr.
Roman Catholic theologian and Bible translator