What about the US, then? We have a form of multiculturalism where minorities are on average oppressed, but integrate rather than lash out (usually).
I'm not saying we're a shining model or anything, but we have a ton of different cultures living together more or less harmoniously, even taking into account the habitual blacks vs whites squabble.
Humanity also has a pretty extensive history of conflict, even in more mono-cultural societies. Hell, you could take any mass of people, all from the same background, race, ethnicity, religion, etc (take your pick), put them on an island for a hundred years or two, and you'll get factions.
You'll get people fighting over how they think the place ought to be run. You'll get people fighting over who gets what and why these guys have all this stuff and we have less stuff. If nothing else, it comes down to availability of resources and numbers of people (along the lines of Dunbar's number and all that).
Sure, ethnicity, race, and religion are convenient signifiers of groups of people and usually when you have two or more groups with significantly different cultures, you've probably got two or more groups with some level of inequality and power imbalance so you're already halfway there. But even if you don't start out with cultural differences, they'll pop up eventually. Groups of people over a certain size will always have a harder time avoiding conflict when things get tough. It's almost in our nature to split into "us" and "them" just to manage it all.
Europe is currently pretty much ordered itself into nation states. It required some bleeding over the centuries. If you look at Swiss and Germans, they can hate each other and be internally fine. Both are using 2% of their GDP in defense to fend of each other proactively. There is balance of power, which result very little violence.
Now if both countries would import so much people from middle east that they would constitute something like 30% of the population of each country. Now ethnic Germans and Swiss seem to be "the same group" compared to Muslims. If shit hit's the fan, there is new pressure to readjust the borders. And move the people. Usually that stuff causes bleeding.
You could say that "if" is completely unrealistic right now. I agree. Nothing that has happened in Europe during past 15 years really warrants any policy change regarding immigration. The point was largely theoretical, just like your island.
I'm not saying we're a shining model or anything, but we have a ton of different cultures living together more or less harmoniously, even taking into account the habitual blacks vs whites squabble.