I want to buy an Apple Gift card for a relative in Italy. I reside in the US and have a US credit card
For example, could I call an Apple store in italy and give them my credit card over the phone?
There is no easy way to accomplish what you want. There are various reasons (taxes, government regulations, exchange rates etc) that you must purchase the gift card in the country or region it will be used. An Apple Gift Card purchased in the US will not work in Italy. Sorry.