The US has less ground to stand on [than Europe to deny immigrants] considering our country is what it is today (whether that is a good thing or not, is up for debate) because of immigrants. Of course Ellis Island has been closed for quite some time now, and things have certainly changed.
To be clear, I'm making no argument in either direction. I'm simply saying a country founded by immigrants and made up of immigrants deciding to stop accepting immigrants is a significant change in philosophy. It is much the same but in inverse of what countries like Sweden, Germany, England, France and others in Europe are doing.
If I was to make an argument, I'd say the cultural changes taking place in Europe because of immigration is probably correlated to the increase of crime, and the fact the US is made up of so many different cultures that it is probably a reason why the US traditionally has more violent crime...different cultures clash. But that doesn't sit well with people who champion multiculturalism who also tend to place the blame on gun owners...