I certainly don't think there is anything wrong with being proud of your heritage. But would you say you were Italian if an Italian person who was born and raised in Italy asked you where you were from, or would you say you were American, but your family is from Italy? Or that you are an Italian American? I guess the point is, if you move over here with an American accent, but tell people you are Italian, it seems it would just confuse them without a lengthy explanation of how you are Italian and American. Same applies if you are Irish-American, but are living in Ireland, then you really can't say you are Irish, necessarily, because you don't have the same accent as they do, and the culture you grew up in may be similar in some ways, but different in others.
When in the UK, I would most definitely say I am American. But when here in the states and I tell people my last name, they look at me and say, Oh you must be Italian. My family lives and breathes our Italian Culture.
I think when people start throwing out 1/8 percentages, that is a bit silly but to each their own. I am Italian-American. Where I live in Tampa, we have a vast and rich culture of Hispanics and Italians reaching back generations. Our cities founding industry built upon the trades that our Italian-Cuban-Spanish relatives brought from our home countries. To dimiss this.... is ignorant.
In our family we celebrate in the ways we are most familiar with Italian traditions. Food, music, large gatherings when people are spilling out of the house because there are too many of us.
We recongise and celebrate the freedoms that America (sorry) The United States has provided us, especially on days like the 4th of July whatever, it is then we roll out the Bar-B-Q to cook hamburgers and hotdogs. But on all other days we use our Bar-B-Q grill to cook our Italian Sausage

and grill our onions and peppers.

Not sure why this is sooooo controversal, why can't we just be tolerant of our differences. I know once I explained it to my BF he came back and apologised for not understanding, he certainly didn't argue with me about it, even though we both have different views on the topic.