This topic has actually been discussed for several times, but it's been a while...
That's an interesting question.
If they managed to capture, occupy and otherwise integrate it (well, as far as possible) then I expect that their will be constant crusades
for a few centuries until either it's retaken or the Christian states basically get their asses collectively handed to them to such an extreme
as to stop.
I’d guess a Crusade to free Holy Rome and a campaign of Reconquista in the Italian Peninsula like the one in Iberia.
Much like when the Saracens invaded Italy and Sicily.
While there will going to be crusades, it won't be so much of an annual thing. IOTL when the news of Ottoman troops have landed in Italy reached Rome, the Pope decided to pack and leave for Avignon. Seemingly related to that, France actually took the invasion with very little respond. Had Italy been conquered by Ottomans, I think there will be two vital consequences :
1) Ottomans setting their own client Pope in Rome (or keeping the Pope in Constantinople instead), and..
2) The Pope in Avignon would come under the dominance of the France.
Both conditions seem ripe for igniting a colossal splintering of Roman Catholic Church. Not to mention the roots for Reformation in Germany were already there back then. The whole Italy would most likely come under he umbrella of Ottoman-backed Pope, with the possible exception of Piedmont (if it'll be swallowed by France instead), while French dominance over the "free"-Pope will bring political consequences that will effect Europe at large. Some may argue the splinter of the Church won't be inevitable, and they may have a point, but I think the Church's unity will be in a very dire situation....
As for Ottoman Italy, I think that it can last for centuries, and will be generally fairly stable throughout the period. I don't see the Italian states north of Rome will have the will to form a solid coalition opposing invading Ottoman troops, which were generally considered as a horde of invisible monsters at the time, and I can see most of them will eventually become vassals of the empire. Besides, being under empire's umbrella will mean for the Italian traders that the eastern trade will be opened up again for them....
Also, with Italy under Ottoman domination, the Meds will be almost of an Ottoman lake, if not already one. Emirate of Granada can and will may utilize this situation to ensure their survival against the onslaught of reconquista....