Immigration and Generation - New Jersey Housewives take trip to Italy

In the Real Housewives of New Jersey, we see where all the housewives with the exception of Danielle, all went to Italy for a vacation. The Manzo's took their parents, so too did Jacqueline. Teresa speaks Italian along with her husband who was born in Italy but moved to the US at age 1. Teresa and her husband are the closest to their roots in Italy compared to the other housewives. It was interesting to see how happy she was to take he friends there. Her family seemed loving also was he husband's family. They all greeted each other with kisses to the point where for me it was overwhelming. I liked it though. In one scene, Caroline talked about how weird it was for her to see Teresa and her husband being so "Italian" for lack of a better word. She continued to make the point that her grandparents left Italy to become Americans. It gave the impression that in the US she is all Italian American, and in Italy, she is all American. One may begin to wonder if she was could not appreciate her roots. My question is, as the US continue to be the number one destination of new immigrants, how will the off spring of those immigrants, whether 1st or 3rd generation refer to themselves as - Americans or American? Give me your thoughts!!