It was really not until the founding of Greece itself in 1821, and the eventual settlement in the 1830s, that those Greeks who now lived in "Greece", called themselves "Greek". People who spoke Greek, and still lived in the Ottoman Empire (where most Greeks continued to live until post-WW1), would have continued to call themselves Romaioi (Roman), or more specifically Christians (along with everyone else in the Balkans). You have to remember that the Ottomans called the Balkans: Rumeli, meaning Rome.

However for many Greeks, living within Greece, the idea of a "Greater Greece" (Megali Idea), in restoring the Roman Empire, was still a live dream and aspiration, up until 1923, when they were finally driven out of Turkey, by Mustafa Kemal "Ataturk" and the Turkish nationalists. After that, there were the only large scale population transfers, post-WW1, between "Greek" Turks/Muslims (including Albanians), and "Turkish" Greeks. After that level of homogenization, any aspiration of restoring the Roman Empire, died with the ethnic cleansing, and so did any pretense to the identity of being Roman.

Today in the area of Trabzon (Trebizond), there are communities of Pontic (Old) Greek speakers, who still consider themselves to be Romaioi.

If you want me to expand on any given topic, just ask.

Source: The Balkans: A Short History - Mark Mazower

If you want other sources I can also provide them, Mazower's book is just the most recent one I read that mentioned this issue.

Follow up question, at what point did Hellenes stop considering themselves Hellenes and start considering themselves Roman?

As others have said, the development of a separate Greek identity did not truly occur until the creation of the modern Greek state. This is illustrated well by an anecdote told by Peter Charanis, born on the island of Lemnos in 1908 and later a professor of Byzantine history at Rutgers University:

When the island [of Lemnos] was occupied by the Greek navy [in 1912], Greek soldiers were sent to the villages and stationed themselves in the public squares. Some of us children ran to see what these Greek soldiers, these Hellenes, looked like. ‘‘What are you looking at?’’ one of them asked. ‘‘At Hellenes,’’ we replied. ‘‘Are you not Hellenes yourselves?’’ he retorted. ‘‘No, we are Romans."