I just got back from Rome last week. We had to show our vaccine cards to get inside every place except stores and you could eat outdoors at restaurants without showing your card. Masks are mandatory indoors and while we were there they added an outdoor mask mandate for certain busier areas. My daughter is studying there and just shows a picture of her card on her phone but I carried my card with me.
The only time we had any question about the US vaccine card was at a nicer restaurant called AdHoc when the server was worried that we didn’t have the Italian pass. We explained to her that Italy does not issue them to visitors and she ended up being fine with it. I was surprised she questioned it because I know other tourists go there (I found out about it on Trip Advisor) so I am sure she has seen the white cards before. I’m wondering if she was just worried because the new rules started up while we were there so maybe she was being extra strict.