WI: Papal States unites Italy?

I want to ask a question.
Could Papal States unite Italy when it does have a peak chance some time after HRE's ouster of Italy? I expect that Papal States would focus on annexing northern Italy first before persuading Spain to give her territories into Papal control himself. Is that possible? I would like to know. I personally think that if most of Italy is under HRE, Frankish, or Lombard i don't think Papal States might have a chance of uniting Italy or else it might be something unknown would happen to it.
If it could, how Papal Italy would work in inter-European relations and how massive butterfly effects will be?
 
As I see it, the best time period would be after the end of the Hohenstaufens 1268 but before the Italian Wars of 1494. This to me seems the time in which Northern Italy was least under Imperial hegemony but not yet within the strong foreign interests of France and the Habsburgs. The only problem is that Aragon starts acquiring Southern Italy in 1282 so there's an even smaller window for the whole peninsula.

You could have the Pope try to solidify the Lombard League into a continuing and increasingly centralised permanent structure and not just an alliance of convenience. Under the right conditions this could allow the Pope to become a shadow ruler of Italy. The problem is that the moment you declare a formal Italian state you start to have significant opposition from the HRE regardless of how long it's been since they lost de facto control.

Not to mention that the secular nobles would resent responding to a religious overlord, so I doubt if a Papal Italy is an enduring state.
 
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