It's a valuable discussion to have, but as per usual there's this American mea culpa about race that's just so fucking tedious.
In the late nineteenth and early twentieth centuries, America led the world in race-based lawmaking, as a broad political consensus favored safeguarding the historically white character of the country.
This is due in no small part to the fact that America was an ethnically diverse environment where the prevailing tribe was being threatened by the prosperity of the aforementioned Italians, Irish, blacks and what-not. Spain had long since expelled the Moors, ejected the Jews and exterminated the Protestants. Europe had no immigration to speak of. The Russians had been practicing pogroms for a couple hundred years. Japan had sealed its borders to all foreigners for 200 years, only to be drawn forcefully out of isolation by Commodore Perry and an American armada. When Hitler took note, it wasn't because of laws keeping the Irish down or whatever as the Hapsburg Empire had forbidden Jews from owning property for fuckin' a thousand years. It was because the American empire displaced Native Americans and corralled them on reservations for easy domination and steady attrition.
The Nazis didn't need to follow American law to make miscegenation illegal; such laws were based in European codes going back to before the Puritans. Where the Nazis followed the American lead was in a fascination for eugenics and a desire for racial purity through conscious breeding programs. If we're going to throw Americans under the bus for anything, eugenics is worth throwing us under the bus for.