So Constantinople fell in the 1450's, America was founded in the 1490's, and colonization began soon after. Although it wasn't exactly massive continent-wide settlements, Europeans did begin popping up in America, many fleeing persecution in Europe.
Did any Byzantines wind up there? They'd be quite old, but not within impossibility, and many claimed to still be Byzantines even after the fall of the Empire.
So did any 'Romans' end up in America?
At least one did!
The story of Ferdinand Palaeologus is an interesting one. His father, Theodore Palaeologus, arrived in England from Genoan-controlled Chios via Italy in 1596 claiming, according to his burial plaque, to be a great grandson of the younger brother of the last Byzantine emperor and the heir to the Roman Empire. Theodore was entertained in the court of the Earl of Lincoln before he obtained some property, married, and established a new family in England.
Ferdinand Palaeologus and his two brothers fought for the Royalists during the first part of the English Civil War. After the Royalists were defeated at the battle of Naseby in 1645, Ferdinand fled to Barbados and acquired a 200 acre plantation. There he lived for more than 30 years and also served as warden and vestryman of St. James Parish Church where he was buried in 1678.
While the exact familial relationship between Theodore Palaeologus and the last emperor is the subject of some debate* there doesn't seem to be any reason to doubt Theodore's eastern Roman ancestry. Many Byzantine nobles fled to western Europe during and after the empire's collapse in the 15th century, and Theodore apparently possessed enough money and knowledge to not be outed as a fraud during his 40 years of living in England.
*I haven't found any information on "John, son of Thomas, second brother to Constantine," mentioned on Theodore's tombstone. This may indicate that Theodore was not completely truthful in describing how closely he was related to the emperor, but according to a sparsely sourced blog post Theodore was the head of his branch of the family, and other members of it supposedly arrived in England preceding and following him which would make the idea that he was an outright impostor seem unlikely.
Related question, would they have considered themselves "Roman", "Greek" or "Byzantine"?
Oh yes in fact.
The current holder of the (defunct) title Emperor of Rome, Bayezid Osman, lives in New York City right now. He's 91 and doesn't have any kids, but I'm sure there's enough of his family around to pass the title hereditarily to someone else.
And for those who're not in on the joke (which I don't actually regard as such), the Ottoman Sultans, after conquering the Byzantine Empire, retained the title "Emperor of Rome" as amongst their titles because of their blood ancestry to Alexios Komnenos through his grandson John Tzelepes Komnenos, and of course the right of conquest by holding onto the city of Constantinople.
Mehmed II took his title quite seriously, and was planning to launch an invasion of the city of Rome itself.
