Well, the Roman Empire lingered around in different forms until long after the fall of Rome in 469 AD. The Eastern Roman Empire was around until 1453, and the Holy Roman Empire until 1806. It's hard therefore to pinpoint a moment in time at which Italians stopped calling themselves Roman. read more
Italy before the reunification( 1861), was split in many little states, so the identity of Rome( replaced by the catholic church) faded over the decades after the empire collapse, and the people did not recognize themselves as an united nation. read more
“Greek” was used by Latins as an insult (you find it, for example, in Villehardouin's account of the Fourth Crusade), to diminish the Roman empire to a Greek kingdom. read more