They don't now only because there isn't a lot of it, so it's not a political issue. How many recent German or French immigrants are there in the US? Not enough for anyone to notice. Back when there was large-scale immigration from Europe though, a lot of people minded, and formed parties specifically to agitate against the Irish/Poles/etc.

You see that now in Europe. UKIP isn't complaining only about muslim immigrants (though they complain about them loudest), but also about continental-European immigrants, especially eastern Europeans. I suspect something similar would happen in the U.S. if there were a large influx of European immigrants.