I was going to write a post speculating about some scenarios in which the papacy might have tried this at different times (during the Carolingian period shortly after the Donation of Pepin, during the Investiture Contest and the conflict between Guelphs and Ghibellines, during the early modern period after the Protestant reformation, after the Congress of Vienna, etc), but I think that a shorter answer will suffice:

They didn't have the raw military power to achieve very much against all of Italy, and they would have squandered their spiritual power by adopting a very un-Christlike policy of belligerent expansionism.

Before the Protestant Reformation, the likely outcome would have been a Holy Roman Empire-sponsored antipope becoming regarded by most Catholics (at least in Italy) as the legitimate spiritual authority. After the Reformation, the consequences might have been even more dire -- such actions might lead to the wholesale collapse of the Catholic Church itself.

That's assuming that a pope wanted Italy-wide conquest. Some may have been interested in that. Some were more interested in spiritual than temporal affairs. Some just wanted wine, women, and song. Even for a pope interested in conquest, though, it wasn't a realistic goal.