Trump is having a tantrum because Italy refused to assist with the Strait and even forbid "offensive" flights from the Sigonella airbase in Sicily. Before that, Trump did not consult with Meloni or other European allies before launching the war, with Meloni being the most humiliated of the bunch, since the Italian Defense Minister (Guido Crosetto) was literally vacationing with his family in Dubai on the day the war started.
Meloni has also pissed Trump off in other ways by being a big supporter of Ukraine and NATO; she also had a domestic political need to distance herself from Trump on trade (a pasta tariff was floated as punishment for unfair trading practices). She's a shrewd character who dropped her best friend Orban at the drop of a hat as soon as the political winds shifted.
Culturally, for Trump to say that Meloni was "begging him" for a picture is just a massive insult and loss of face that is not acceptable in an Italian context. She certainly pulled Tajani's trip to Miami. More consequences will follow. Italy has already added new barriers to Italian-Americans getting dual citizenship in recent years. I expect that these will now increase substantially. They will also slow-walk consular functions. They have a lot of little levers, especially on tourism (which is a domestic political problem anyway), to make Americans and Italian-Americans suffer.
Since most Italian-Americans are (unfortunately) conservative Trump voters, there's a hint of justice to it.