The answer is it depends ... from the US Embassy in Italy:
Beginning on May 1 and valid until June 15, 2022, travelers from the United States may enter Italy if they are asymptomatic AND present one of the following:
An Anti-COVID 19 vaccination certificate for an European Medicines Agency (EMA)-recognized vaccine. At present, EMA recognizes the following vaccines: Pfizer-BioNtech, Moderna, Johnson and Johnson, AstraZeneca and Novavax. The date of the last dose of the primary vaccination series must be at least 14 days prior to travel and no more than 270 days before arrival unless a booster shot was administered. Travelers vaccinated in the United States can prove this via the “white card” bearing the CDC logo.
A medical certificate confirming recovery from COVID valid for six months form recovery.
A negative molecular PCR test result carried out within 72 hours of arrival or a rapid antigen test result carried out within 48 hours of arrival
Passengers unable to present a valid vaccination, recovery certificate or negative test result will have to self-isolate on arrival for five days and undergo a molecular PCR or rapid antigen test at the end of the isolation