Also, I will state this as I cannot seem to find the "edit" button; your ethnicity is determined by which specific cultural background that you practice. As an Anthropologist, I figure I should chime in on this subject.
Do you celebrate mostly Celtic holidays and follow Celtic traditions? Even though you were born in America, you may state Irish as your ethnicity. In fact, ethnicity is one of those things that truly is a social construct. No one may state what ethnicity you truly are as it is a practice, not a group of belonging.
I agree; the question asks for ethnicity, but the answers mostly deal with racial classification. Weird.
Paraphrasing one of the most gifted young actresses out there today; "the really amazing this is, it is nobody's goddamn business." (Emma Stone in Easy A.) It becomes unimportant when people stop asking. I was born in the Great American Melting Pot, and raised to not care whether my playmates were Christian, Jewish, British, German, Iranian, Korean, Black, or anything else. For all my life everyone wants to know "what" I am.
Have your little Greek festival, or celebrate Cinco de Mayo. I'll stop by, have some food, maybe even buy a t-shirt with my name in your native language. Just don't ask me what I am or you'll get another lecture from Papa
Papa
P.S. I'm mostly Irish, but born in Hawaii, with a smattering of Scandinavian. (Think a fish eating whiskey drinker who is not confused by Ikea furniture.) My ex-wife is Czech, with some Irish, but was raised "Texas first!" My kids are Hispanic and native Mexican (yes, there is a difference), but they are being raised American.