General Social Survey respondents, both Americans and immigrants in America, were asked: "Some people say that the following things are important for being truly American. Others say they are not important. How important do you think it is to have American ancestry?" I've listed the percent who answered "very important" by ethnic group (sample size = 1,184):
I'm surprised how common it is to believe that you need to have American ancestors to be truly American (I shouldn't be surprised, given that just about every country in the world focuses on ancestry). "Ancestors" sound pretty old, but I take it that it means at least that your parents had to be born here. So it suggests that people who agree don't think that immigrants can be real Americans. I didn't put the numbers in the above table, but 54 percent of all respondents and 52 percent of whites feel it is very or fairly important to have American ancestors to be truly American.
In the table, you can see that old American groups with a history of mistreatment are most likely to think that immigrants cannot be real Americans. Those of English or German ancestry, on the other hand, are more likely believe that Americanness is not an ancestral thing. Groups with many recent immigrants and (pro-immigration) Jews are least likely to place importance on ancestry.