There is significant compatibility between Eastern and Western cultures, and each has a lot to offer the other. Not really the case with Islamic culture - but perhaps one thing it can unwittingly provide is help us to rediscover the values that made the West great.
It's also unfair to compare the current situation with previous cultures that have immigrated. There are no Irish enclaves or no-go zones that I'm aware of. There is plenty of interaction between Chinese and "white" communities.
But in the UK and France, the children of Muslim immigrants are MORE radicalized than their parents. Their majority views on abortion, domestic violence, LGBT rights and democracy are abhorent. There's no ignoring this - multiculturalism in this instance is failing.
You're kind of right in that, although I do believe in Western liberals values, I currently have little confidence in our collective desire to promote them. In our current confusion about what our values are, we are ripe for exploitation by other inferior cultures.
And yes, I said inferior because I'm not a postmodernist or cultural relativist. I may not have all the data, but I'm not afraid to make the value judgement that so many others are.