That's really an open question. Most scholars consider nationalism a modern phenomenon, but some (like Mommsen) did think Rome in the first century CE approximated a nation state. I'm currently writing a dissertation arguing for the something-like-nation-state position, but I don't think nationalism is involved.

Nationalism is an ideology, that I at least don't think extended far beyond the elite. It is unlikely to have occurred in the ancient world, and evidence of it penetrating the broad swath of the population is unlikely to survive. You can't really say what the middle class Roman (or Italian) thought about the nation, because that kind of evidence just doesn't exist.

Rome (and Italy) did possess many institutions that modern states use to construct nations. My go-to guy here is Kymlicka. His book, Politics in the Vernacular sets out a number of institutions that modern states use to construct a nation purposefully. I don't think Rome purposefully propagated a particular culture (dat alliteration), but I think the elements were there and were still active.

The institutions Kymlicka emphasizes are a national assembly, economic integration, national standards for education (among others - that's what came to mind right now). Kymlicka argues that modern liberal nation-building states (his term) employ these institutions on purpose to build a political state that values the individual, not the community.

I don't think Rome did this on purpose like the modern nation-building state does, but I do think these institutions operated on the same principles. Similar processes have been demonstrated in France (Weber).

So to sum up, I don't think Rome was nationalist, but I do think it was something like a nation in the first century CE.