This has kind of been the question of the day, and since the Paris riots, France, one of the biggest advocates of wider diversity has been made to look quite foolish.
Overall I think Europeans and Americans have very different attitudes about this. Europeans I've talked to do feel that nation-states should be based loosely on ethnic homogeny, while Americans feel it's largely not necessary.
I can understand European pessimism about multi-culturalism. You guys have a a long history of ethnically fuelled wars, and not much experience with proper integration with immigration to have much faith in it.
The U.S. is a little different. (I didn't say better. ...Just different.) We have a history of slavery, but then a legacy of rehabilitating the abyssmal racial relations it caused for so long. We have a long history of immigration, and a pretty universal integration into the American population as a whole, within a generation or two. And most ethnicities drop their gripes with each other once they reach here. Turks and Armenians get along, Scotch-Irish and Catholic Irish have no problems, and Germans and German-Jews get along and even frequently marry. ...So maybe you can understand why we're a little more optomistic.
Also, there's a difference between multi-ethnic, and multi-cultural. Multi-ethnicity is fine. Multi-culturalism is ok to a certain extent. It's ok as long as everyone speaks the same common language, and has similar economic opportunities. Our National and now Multi-National corporations, and mass media have provided enough of a common American experience and culture in most parts of the country. Even though it's very large, culturally it is somewhat like the Borg. Even geographical accents have largely disappeared because of popular television. Anything that's too different is an instant subject of a hollywood movie poking fun, scrutinizing, and celebrating all at the same time. (Like Raising Arizona made fun of the South West, and Fargo made fun of the Northern midwest.) ...So we're not multi-cultural, as much as we are multi-ethnic.
Europeans are largely inexperienced with immigration, and as a result have accepted large groups of immigrants from one culture or location, which has created an underclass. ...And in Europe the result has been a large Muslim/North African/Turkish class of people who can't get decent jobs, and have more in common with each other, than their new countries of residence. ...The United States has usually been a lot better at accepting immigrants from all over with the American English speaking culture being a common core that they all fold into. Europe has yet to learn how to do this, and some of the smaller countries will have a harder time doing it, since the trend is to want to speak a global language like English, Spanish, Chinese, or Russian. (...And not Finlandian.)
Also, Europeans have forgotten how to make babies. So you guys have a looming demographic crises. We do a better job of that, and have more inter-racial marriage. I have a family member married to an African-American, and another to a Chinese-American. Do you have any relatives married to any of the new immigrant groups? I've dated seriously outside my own ethnicity. ...Have you?
Europe is going to have a lot of problems no matter what they do, until they truly accept these new immigrants, and have more honest social mobility. Countries like England will be fine, as they are diversifying their immigration and accepting Brasilians, and Carribean immigrants. Countries like France are going to have a harder time, because they have failed to diversify soon enough.
Whew!! ...Long winded post Indian Summer, but hope that answers what some of my thoughts are on it.