The Greek term for 'Roman' as a name for Eastern Orthodox Greeks (Romios / Ρωμιός) is still around in the Greek language and understandable, but considered 'old timey' equivalent almost to the American term 'Yankee' as it would be used for all US people (source : I'm Greek, I can link to modern media such as songs that use the word of you want).

This book apparently has the whole story (https://books.google.gr/books?id=iWs0Lh57NvwC&printsec=frontcover&hl=el&source=gbs_v2_summary_r#v=onepage&q&f=false)

To make a long story short, the modern Greek name for Greeks (Hellene) has consistently meant 'pagan' until the 18th century. The Greek subjects of the byzantine and Ottoman Empire called themselves 'Roman' or Ionian (unan)'. 'Greek' and 'Byzantine' are western inventions popularized after the fall of Constantinople and reloaned to the Greek language after the 18th century again.

So if you asked a person in 1821 (the year of the Greek independence war) what they identify with you had good chance that they could pick any one of : Romios (Roman), Graekos (Greek), Ellinas (Hellene), Unan (Ionian in Turkish) or even just plainly Christian.

Side note: When the Ottomans defeated the Byzantine Empire and conquered Constantinople, Sultan Mehmed II adopted the title 'Kayser-i Rum' - Caesar of Rome. Implying that at least from the Ottoman perspective, the Byzantines continued to be Romans right until the very end. The title remained in use right up to 1922.