See, the problem with that is, pretty much all of Europe and most North Americans, Australians and quite a few other people, have just as much of a claim to ancient Roman culture as you do.
Other than geography, what distinguishes you from any other Italian? I realize that there are a few different ethnic groups recognized in Italy, but, by and large, Italian people are Italian, not Roman. I'm not trying to be snotty here, I'm genuinely curious. My study background is in ethnic conflict, so, I understand the importantance of ethnicity, but, honestly, you're the first person I've ever seen who is claiming that Rome is a separate ethnicity and culture from Italian.
No it's not really about ethnicity. There are plenty of dark skinned people, for instance, that are just as roman as I am. The ancient roman themselves were very ethnically mixed. Italy's geographical position in the middle of the mediterranean sea is very important to consider. We had people coming in from the north (north europe) and from the south (north africa) for millennia.
All those people brought part of their culture, but they were all eventually assimilated, either in the ancient roman culture or, after the fall of the western empire, in the many italian regional cultures.
And regional cultures are the crux of the matter: a lombard is not a tuscan, a tuscan is not a roman, a sicilian is not a naepolitan, a genoese is not a venetian, etc etc.
My regional culture is roman. My native dialect is the roman dialect. If a venetian dude (especially an older fellow) speaks to me in pure venetian dialect, I'm not going to understand a word he's saying. I'd probably understand a spanish guy better, and I never even studied spanish!
So, what is modern roman culture? It's the regional culture of the people of Rome and surrounding areas, tracing its roots to our ancient roman ancestors (753 BC-476 AD), going through the primacy of the Pope and the Roman Catholic Church (edit: latin was and is their official language, I'm going to add) for more than a millennia, until the kingdom of Italy took Rome from the Papacy in 1870 to make it its capital.
Newsflash: this last development is relatively recent history. While modern romans certainly consider themselves italians, if you ask romans whether they feel more italian or more roman, the answer might surprise you.