I don't think so. I interpret paradoxes as being either philosophical impasses (ie, 2 conceptually true statements conflict each other in a way that makes you question where one statement's truth ends and the other statement's truth begins) or a situation in which a solution is unintuitive.

A Catch-22 is more of a physical and intentional impasse, where obstacles are intentionally set up in such a way that people are unable to make a choice. For instance, in the original example of a Catch-22, there is no philosophical argument saying that only insane people are allowed to not fly - it is an arbitrary rule that some higher-up established. And likewise, it is entirely arbitrary to define insane as being willing to fly.

I guess to simplify my stance, it's a paradox if it makes you think "the universe has made this unsolvable" and it's a Catch-22 if it makes you think "some asshole made this unsolvable"