Tim,
 
While I'm not an expert on Byzantium, I believe it was always a very Greek place. It held up to the Moslem onslaught for quite a while but fell to the Ottoman Turks in 1453. The Eastern Roman Empire, centered in Byzantium, fell around 788, I believe. I don't believe it ever held sway over Syria or Arabia, thus Byzantium was never Islamic. The Ottoman Turk empire, however, was of course very Islamic. If I am wrong, I'm sure someone will correct me here.
 
The western leg of the Roman Empire never really died. It retracted and came to rest in the Vatican, which I believe sits over the former palaces of the Roman Emperors. So, where Byzantium stopped being Roman rather early on, the Western Leg persists and its former political constituents remain as modern nations.
 
I still must hold to my belief that conventional wisdom about the Beast rising out of a Roman milieu is correct.
 
Your brother in Christ,
Matt in Hayward, CA