I think it really does depend on who unites Italy. If Venice or Genoa do it (very unlikely of course, don't know why they would go through all that trouble) then we should expect Eastern Med interests to be considered part and parcel of the united Italy's interests. If it's the Papal States then they might be more interested in European religious matters, like a second Habsburg Spain.
However, the defensive imperatives of a united Italy (especially considering that as a peninsula, there is a lot to defend) probably are the same for any united Italy no matter the government it is under, and these imperatives likely include controlling the Alpine passes, the Adriatic coast, the Malta Channel, Corsica and Sardinia, and Tunisia. Greece is probably required if Italy is to have any Eastern Mediterranean aspirations.
This means that a united Italy is almost guaranteed to cause trouble for the Byzantine Empire and the North African sultanates/caliphates, as well as being a potent rival to the Spanish at sea. On land Italy might have to play a fine diplomatic game between France and Germany/Austria, while extending influence in the Balkans in order to create a 'buffer zone' for its Adriatic holdings. Irredentism will always be an issue because of the legacy of the Romans.