In Modern Turkish, Rum (pronounced like the English word "room") still means Greek. However, this term only refers to the Greeks of Turkey and of Cyprus; the Greeks of Greece are "Yunan" (from the same root as Ionian). So yes, the Ottomans called them Romans, but that's also how many of the Greeks called themselves.

After all, by that point, the Greeks in Greece and Asia Minor had been Romans (what we call Byzantines, but they called Romans) for more than 1,000 years. The Greeks generally called themselves Romans until nationalism took hold in the 19th century and there was a big debate within Greek society whether they were "Romans" or "Helenes". Here's an old post by /u/Blancbard:

I think it is worth mentioning that the name Hellen in the Greek-speaking world fell out of use and it got replaced by the term Roman. Especially after the Christianization of the Empire, "Hellen" became associated with paganism and later it fell pretty much completely out of use (as far as we know, the last ones who used it were the Maniots who were still pagan up until the 9th century, as the Emperor Constantine VII Porphyrogennetos mentions in "De Administrando Imperio").

Until around the formation of the Greek state in 1830, Hellen was essentially never used, except some times romantically. Greeks were self-identifying as Ρωμιοί, or, more formally, Ρωμαίοι (both meaning Roman) and their language Ρωμέικα. This kept being the main name of the Greeks that were outside Greece until well into the 20th century. Today, Ρωμιός is still used sometimes in traditional songs, and in various other contexts.

There's a Greek dialect spoken in Turkey by a small Muslkm population on the Black Sea in the district of Of between the cities of Rize and Trabzon. You can read more about it here, but it's relevant because they call their langauge "Romeyka".

The question of how one exactly identifies is an important one for nationalist movements. As Germany was uniting, were you Bavarian or Austrian, or were you German? Were you Tuscan or Sicilian, or were you Italian? Were you Bretagne or Provençal or Alsatian, or were you French? In most of those cases, the question is between a sub-national identity and a national one. The Greek case was similar, except it wasn't between a national and a subnational identity, but between one that was part of the Ottoman imperial fabric or one that sought to go its own way. This was only really being crystallized in the period you're asking about.

Here's a slightly longer post I have about the emergence of a separate Hellene identity, whereas before it had just been a synonym of Roman. Make sure to read down the the thread to my follow up comment.

But, in short, yes, and in some contexts Turks still call Greeks Romans. See also the very in-depth Wikipedia article Names of the Greeks.