Better question: Would the Spanish people and government be okay with an Italian ruling as their king?
If they came to Spain, learned the language and abided by the laws and customs?
Maybe. Its not like the Carlists arent pracically foreigners by now. For that matter the Bourbons had their origins in France.
Will Franco choose to restore the monarchy as per OTL? And with Otto now ruling Hungary, does that mean that he could turn to Italy for a non-Bourbon (re: Carlist) candidate?
However Vittorio Emanuele III had other projects. He felt he didn’t have long to live and he yearned for a final achievement to cement his legacy is to make the Savoia the premier European dynasty, by installing parents on vacant or newly created thrones. He did achieve an important result in making his nephew Aimone of Savoia-Aosta King of Croatia, with the name of Tomislav II. It was the price Pavelic paid to Mussolini for Croatia’s freedom and the annexation of Bosnia. This emboldened the king, who plotted to make Aimone’s older brother, Amedeo, the current viceroy of the Africa Orientale Italiana, ruler of a most prestigious government. In this project was supported by his son, Umberto, who apparently wanted Amedeo out of Italy for jealousy issues, the viceroy being much popular than him and above all being suspected to be too near to his wife Maria José. Naturally Vittorio Emanuele looked mostly to Spain – Amedeo’s great father was king for three scarce years and failed miserably, but the nephew was considered much more adept to become King. But Mussolini wasn’t interested in forcing the hand of Franco over this, both in truth preferring Juan Carlos of Bourbon, who was in Rome at the time.
Spain: Become the number two power in the Bloc once rebuilt from the Civil War.
Bulgaria: Not getting steamrolled by the Soviets.
Turkey: Re-establishing Turkey's eminence, particularly in the Middle East.
Portugal: Establish a secure foreign Empire.
Italy: Becoming the world's third power ahead of Britain, with enough military power to project itself almost anywhere.
Italy has a long way to go, then. The most difficult part is not to get the industries, is to teach and train the hands and minds of men: workes, middle-management, leadership, etc. And Italy desperately need teachers. One of the aspects of the school reform was to limit the number of people who could graduate from university through the harshment of tests. This led to an higher preparation of the graduates, but also an insufficient numbers of them. Also, many professors were dismissed for not swearing fidelty to fascism. Mussolini better soften this measure (like, swearing fealty to not oppose Italy, a more neutral stance). And let's not forget, to open higher instruction to women. alright, Mussolini will probably never do something like that. We need some eminent figure to compel him to do so. Like, maybe Umberto II's wife, Marié Jose of Belgium. She doesn't seem the type of woman without initiave and boldness.
All around the world, we see these conflicts with terrorists trying to take on great powers: in France with Corsica and their Algerian holdouts, in Britain with Northern Ireland and Kurdistan with their Arab towns and regions.
The new kings Belgian wife strikes me as a decent fig leaf for closer ties with the French and British.
ITTL, Turkey may be willing to split off a puppet state which can in turn eat pieces of neighbouring nations to contain all the Kurdish lands - this indirectly increases Turkish power and if there is hostility between Turkey and the Arabs, this gives those a nice target which to hit instead of Ankara or Kostantinyye.
OTL, Turkey will only lose land should Kurdistan come into being.
Speaking of Belgium, I wonder if the massacre of Marcinelle would happen. I think the prelude (immigration treaty between Italy and Belgium of 1946) may still happen because the country needed manpower and well Rome for commercial benefits could concede them. I don't see so many Italians to move as OTL, of course. But enough to go work into the mines maybe.