They did refer to themselves as “Roman”, but in Greek - “Rhomaioi.” From their perspective they were simply the eastern, Greek-speaking part of the Roman Empire.

The Arabs and Persians called them “Rum”, “Rome”, for the same reason - from their perspective the empire never “fell”, the eastern empire was always the same Roman Empire they had always known. When the Seljuk Turks arrived in the 11th century, they borrowed the name used by the Arabs and Persians. The offshoot of the Seljuks that settled in Anatolia called themselves the “Sultanate of Rum”, because as far as they were concerned, Anatolia was the Roman Empire. The poet Rumi’s name, for example, literally means “the Roman”, since he lived in Seljuk Rum. The Ottoman Turks who replaced the Seljuks continued to use the term “Rum” to refer to all Greek-speaking Christians, right up until the 19th century.

In western Europe, the eastern empire was the Roman Empire until the 8th century, when for various reasons the Pope in Rome no longer considered the emperor in Constantinople legitimate. The emperor Constantine VI had been overthrown by his mother Irene, but I’m not sure why the west was so concerned about that particular coup and not any of the previous ones - presumably because now the empire was led by a woman, but hopefully an expert in that period can weigh in on the specifics. In any case, Charlemagne was crowned as the “true” Roman emperor, and afterwards the western church and secular leaders began to refer to the east as the “Greek empire”.

That name spread to other parts of Europe, so when the Vikings, for example, came into contact with Byzantium, they already knew it as “Grikkland” (and they called Constantinople “Miklagard”, “the big city”). During the crusades, westerners typically called the Greek empire, or the “empire of Constantinople”.

There was an incident during the Third Crusade in 1190 where the western emperor, Frederick I, tried to meet with the eastern emperor, Isaac II. They both claimed to be the Roman emperor, but Isaac called Frederick the “emperor of the Germans” and Frederick called Isaac the “emperor of the Greeks”, and they were both so offended that they never ended up meeting in person.

On very rare occasions when they weren’t trying to be dismissive or offensive, they knew the Byzantines called it “Romania” and used that term. “Romania” was also one of the names used by the western Latin empire that conquered Constantinole in the Fourth Crusade - so they definitely always knew it was called that, even when they purposely chose to call it “Greece”.

The west continued to call the east “Greek” right up until the fall of the empire in the 15th century.

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EDIT: oops, Antiochene pointed to an older answer as I was typing this! Probably a better answer too!

I've answered most of this question here