The Papacy will always come back, although I'm not sure about the time frame, but the usual rules of happiness percentages don't apply. Papal reemergences can be pretty harsh as well, so I usually leave them for the last faction standing in my path. The time-honored way of dealing with the Pope is to take the Papal States, strip it of just about everything, and then abandon it to be retaken. Then take the rest of their territories leaving the Pope with one province, no cash, and an army upkeep large enough to keep him broke and unable to build/recruit anything for the rest of the game.