It isn’t a matter of Italian law unless you personally swear allegiance to Italy and renounce your American Citizenship. If you have not done so, then you haven’t.
Neither of which are required for me to obtain an Italian passport. Italy already considers me a citizen (although they don't know I exist), so there's no need for me to swear allegiance. And the US doesn't require me to renounce my citizenship if I apply for an Italian passport. It appears, then, that I can be both a Natural Born Citizen of the US and a passport-carrying citizen of another country.