This is ASB as hell, but a Europe united under the rule of the pope was something I've always thought would be interesting.
It's not ASB. There's a very small window for it to happen, and it would require an immense amount of luck, judgement and forethought, but during the Middle Ages the Papacy publicly acknowledged themselves to be the overlords of Europe and sought to make the Kings of Europe recognise them as their temporal rulers as well as spiritual. Course, most of the time the rest of Europe just laughed off this claim, but there was one time where one HREmperor (I think it was Otto II, but I'm not sure?) was forced to publicly walk bare-foot into Rome (a sign of shame and penance) to seek the Pope's forgiveness, and at the time, the Pope took the opportunity to force the HREmperor to swear homage and recognise the Pope as the legal lord of the HRE. It went to pot and was forgotten about within a decade or so, but if events had fallen into place it could have been the domino which started off the Pope actually having the influence and power to force the rest of Christendom to actually come to Rome to pay homage in order to receive their crowns.
The chances are incredibly slim, though.
Is it possible for the "states" to be turned into Bishoprics of some kind, the leaders elected by the local clergy, but, for lack of a better term, swear fealty to the Papacy?
The swearing fealty to the Pope? Yes. John (he of Robin Hood fame) swore fealty to the Pope as a legalistic "get out of jail card" move when rebels were attempting to overthrow him, but then promptly ignored that he had ever done it as soon as England was stabilised again. And as above, one HREmperor was forced to swear fealty in order to get his crown back.
Turning the rulers into the puppets of bishops? I can't see it. More rather, in this scenario, the Kings would set themselves up as the Pope's enforcers, and the fact that they had an army each would be enough to keep the bishops from dominating them.