My grandfather, who is now in his mid 90s, came over to Jersey City from Eastern Europe when he was a young boy. If he'd stayed in Russia, his town would have been overrun within a week of Operation Barbarossa starting and he and his family would have been killed. Here, he raised a family and did very well for himself, growing up speaking English and working with all types of people.
You'd expect him to be as grateful to be an American as I am, given this record, but when I talked about it with him a few months ago he sounded like he was still a guest in this country and he didn't particularly feel like an American. Identifying with America, thinking about what it stands for etc. just wasn't relevant to him. Of course, he isn't Russian either, nor is he religious. He just is.
As for assimilation, I agree with Clemenza for the most part. I'm fully aware that most ethnicities still assimilate within two or three generations. My concern is with ethnic ghettos that get large enough that they can drop out of American society indefinitely. You see that with Muslim immigrant neighborhoods in Europe and I think it's been happening in some Hispanic neighborhoods in the US, and I'm concerned about children who are taught in school in a language other than English so they never have any need to learn it.