Gödel was motivated by convictions about meta-mathematics, motivated by his Catholic fatih. Perhaps a purely mechanistic world view would have eventually generated the same result, e.g. Turing's halting problem, but who knows.
Gödel's background was Lutheran, not Catholic, and I don't think he was particularly orthodox. (I have no idea how his religious convictions influenced his metamathematical ones; you might well be right about that.)
Thanks for the correction. I'm sure I've heard biographers (Jessica Goldestein perhaps?) describe his faith as having had a strong influence, but I must have misconstrued it quite a bit as memory faded. You're right, he was raised Lutheran. But his father was Catholic.