That's an interesting question. The people of the old byzantine empire considered themselves as romans, despite being ethnically and culturally greek.
A similar situation happened in east francia, as Otto, supported by the church, claimed the title of Holy Roman Emperor, based in the idea that his power derived from the old roman emperors.
In this sense, "Roman" was, and still is, a title of extreme prestige, which could be given to an empire that continued, to some extent, the tradition of the old lords of europe.
However loosely connected the byzantines, the germans and the turko-greeks were to the old romans, they had something in common: they were christians. Though christianity was a foreign cult to the romans at first, they embraced it as their official religion, thanks to Justinian. The church, after the fall of the WRE, became the central element of power legitimacy across all Europe during the middle ages.
A ruler aspiring for the title of Roman Emperor could be from any place, but he had to be christian. A foreign religion like islam could never be the Imperial Cult, perhaps even more so than others, as it directly antagonizes the tradition of the old emperors. Therefore, a muslim ruler would never be considered roman, not even by his subjects. His empire could be as powerful and influential as rome, but it would be something else entirely.
I wonder what would be the situation of islam today, had the turks been muslim. As we know, all of north africa, egypt and the levant was returned to the christian faith during the rule of the NBE, leaving islam confined to arabia, persia, india and east africa.