The Papal States is a horde faction without a family tree - ie. indestructible in M2TW terms (in regards to the game engine)

The only way you can sort of destroy them is if you:

1. Kill every single Catholic nation completely.
2. Destroy all Papal States armies, agents, take all their towns.
3. Keep killing the Popes until no more can be spawned due to lack of new Cardinals.

The only time the Papal States will have anything to do from this point on is if Inquisitors spawn for them.

Naturally this can't be done if you yourself are Catholic unless you don't recruit any priests yourself. But it's better if you're not Catholic yourself.