The existence and continued renewal of historical enclaves is not an indication that America does not integrate immigrants or that America has not been enriched by their presence. (It's almost absurd to say. America is far from being "little England", and the culture I grew up with was so clearly influenced by the various peoples who settled the area, I don't even know what you could possibly mean.)
America is currently home to almost 50 million immigrants, representing about 13% of our population. This is more than any country on Earth by number, and something like 8th by percentage. This number is just immigrants, not the American children of immigrants.
The population of Chinatown in SF is around 35,000. Only 80% of that is some kind of Asian. The Chinese-American population of San Francisco is around 175,000 (out of around 850,000; total Asian-American population is around 280,000). There are areas of concentration, but you've a great chance of having a Chinese-American neighbor or two wherever you live in the city. The population of Little Italy in Manhattan is just under half white, with less than 10% being ethnically Italian. Similar demographic changes for Little Russia in SF. With the exception of some religious enclaves (e.g. Hasidic Jews), this is probably the case for most enclaves in the US older than (say) 50 years.
Enclaves existed in Europe historically and still exist today. Both in the US and in Europe, enclaves formed both naturally as a thing immigrants did to survive in their new host country and as part of government plans.
Ethnically, in the US the lines are essentially gone. Almost no one cares if your grandmother was Irish or your cousin married a Greek. (There are some pockets where people do care, but they're usually associated with historical beef—or generalized racism—and they're by far the exception.) To the extent non-white people care about ethnicity, it is probably correlated with how recently their ancestors immigrated, with recent immigrants themselves most likely to hold such views. I doubt this is that different from Europe.
We do tend to divide more along racial lines; especially between certain groups—usually with outspoken people in BOTH groups having a problem with mixing. This is seen both socially and geographically; especially in some places. (Even this has changed significantly over my lifetime in places I thought it wouldn't.) I don't really know how this compares to Europe.