No, therefore he isn't
impeccable, which is not claimed. He is able to teach
infallibly in certain restricted situations, just as an Ecumenical Council may teach infallibly -- that is, without error.
We believe the Church is
indefectible -- that "the gates of hell shall not prevail against it" (Mt 16:18). We also believe that an Ecumenical Council or the pope in certain cases can bind the Church to believe something
irreformably as a dogmatic truth or definitive moral teaching (cf. Mt 18:18). If we were bound to believe an error (say, had Arianism prevailed and a Council declared that Jesus is the "firstborn of all creation" but not God Himself) then the Church would have fallen into error and hell would have prevailed against it. The same is true for those things which have not been formally defined but have been believed "everywhere, always, by all." (
St. Vincent of Lerins, "Commonitory", Ch. 2, para. 6), if these things were in error then the Church would have fallen to the Enemy.
That's it. Infallibility does not say that nobody in the Church sins or may say or do things that are really stupid or wrong (yes, even the pope). It doesn't even say that the true faith will not be eclipsed, as we saw in the Arian crisis or the present modernist crisis. Infallibility simply says that the Church as a whole will not fall into error and therefore cease to be the Church but will maintain the true faith (on a dogmatic, not a personal, level) until Christ comes again in glory. This is also why the Church is still the Church, even after Vatican II -- despite the rhetoric of rupture, nothing was dogmatically defined contrary to previously defined dogma (the teachings must only be understood in a
continuity with previous teachings, they do not replace them simply because it is newer).
The pope is a sinner and goes to Confession like everybody else. That has no bearing on his ability to teach the dogmas of the faith infallibly.