The difference is that the ethnic diversity in Europe looks different than it does in the US: in Europe, it's because of immigration. Folks generally don't cross an ocean to then shit on the society they've come to join. Sure enough there's really not that much violence in most immigrant communities in the US. I used to live in the slums of Baltimore, and finally moved to a new apartment in DC. I got to the laundry room and saw a lady in there speaking Spanish to her daughter, and thought "Alrighty, if the immigrants have come here, this is a decent place." I was right.
But the highest crime rate in the US is in the black enclaves in the inner cities. That population was never an immigrant community; it's the descendants of former slaves. We (the American whites) did horrible things to them, and then after emancipation continued to do horrible things to them in part of the country while not really doing enough to facilitate the integration of the liberated slaves and their descendants into society. By the time we passed the Civil Rights Act there were endemic social problems in the US black community, to the point that there's a long and very respectful Department of Labor study into them (the Moynihan Report).
So now in the US those black enclaves have a sky-high murder rate, and the rest of the country has a pretty low one (broadly similar to Europe's). Why? A whole constellation of historical and cultural reasons, many of them traceable back to horrid racism years ago. Should we still blame whitey for the problems? Is it slavery's fault that kids in the ghetto kill each other for silly reasons and don't want to learn to read and write? I dunno.
But saying simply "Europe has diversity too" misses the point: the non-white folks in Europe are there because they came there and wanted to be European, for the most part. (This is pretty similar to Asian-Americans, a group with a low crime rate.) That has vastly different cultural effects than hauling people's ancestors over in chains and wrecking their society.