We learnt in history that after the initial split of the Roman Empire, the east Calle themselves romans, and did even after the fall of the west. Was there ever a point where this stopped?
They never stopped.
To the end of the Roman state it's inhabitants referred to themselves as, "Rhomaioi" and to their state as "Basileia ton Rhomaion" (Roman Empire). Hellenism never saw a serious revival in Byzantine consciousness, especially outside of the upper class. The forums of the Roman Republic, like the Senate and Consulship remained intact almost to 1453.
As a side note, the Byzantinist Peter Charais, who came from Lemnos, remarked that (I'm paraphrasing a bit here)
"when the Island was retaken, the soldiers were stationed in all the villages. Some children came up to gawk at the soldiers. The soldiers asked, "What are you looking at"? The children replied, "Hellenes". The garrison replied, "Are you not also Hellenes"? At this the children replied, "No, we are Romans."
This identity was in use all the way to the end of the Ottoman Empire. Romioi is still a Greek poetic term used for themselves.