No. Several European countries have had significant foreign-born populations for a long time.
https://www.statista.com/statistics/549292/foreign-born-popu...
However, a person from a neighbouring country (e.g. Mexico) would count as an immigrant in the US.
Just because states are massive in the US doesn't make them equivalent to different countries in Europe, which all have their own language, ethnic background, and culture. A German person moving 350 miles to Italy (from Munich to Venice) is definitely an immigrant, with a completely different background, even though the distance itself isn't even enough to go between Austin and Houston in Texas.
It's no longer the case (I read today that 1 in 4 Swedish schoolchidren wasn't born in Sweden!), and I wonder very much what that will mean for the future of Europe. The mass influx of Germans, Irish, Scandinavians & Italians in the 19th century radically changed what it meant to be an American, and it seems to me that Europe is about to go through an even larger-scale change.