What a nonsensical thing to say. I suppose a trip to Chinatown and Little Italy must fill you with fury. You have no idea under what circumstances this woman's family came to the US. Quite likely she was born a citizen, in which case you haven't the right to question her thus.
My paternal grandmother's family became American in an interesting manner. They were Mexicans, living in the Nueces valley, when Zachary Taylor's army came through on its way to thrash Santa Anna's army. My direct ancestor, despite being Mexican, joined up with the US forces, fought, and was killed in the campaign. His family then became US citizens upon the Treaty of Guadalupe Hidalgo in 1849. Have they ceased to identify as ethnic Mexicans since then? No. Have they identified as Americans since? Yes, and see post no.25 for proof. They've been Americans since long before the great immigrant influx from which the majority of Americans can claim descent.
Really, you ought to consider this. Given that "American culture" is, at its base, a unique amalgam of foreign (mostly European and specifically British) cultures, your statement makes little sense.