Way too far removed from the original Roman Empire. The Ottomans conquered plenty of Roman territory yet they weren't Romans. Only the Greek Orthodox Byzantines were literal remnants of Rome, and that is why, for example, Catholic Wales cannot form the Empire.
Actually, as soon as they conquered Constantinople, the Ottomans declared themselves the successors of the Eastern Empire. And soon afterwards, heirs of the *whole* Roman Empire, laying claim to the entirety of the territory that was part of the Empire of the Caesars (the Ottoman Sultan held the title 'Kaiser-i-Rum', which means 'Caesar of Rome'). Which is also why they kept the name 'Constantinople' right to the end - Istambul was the name given in 1923 by the Turkish nationalist Atatürk.
Their claims were not so easily dismissed. When the Ottomans conquered Otranto in Southern Italy in 1481, it was ostensibly to start the reconquest of the Western part of the Empire. And in the taverns of Italy, the talk of the time was that the Roman Empire was about to be reborn, but under a new religion.
So, in effect, the Ottomans were pretty much playing LoR. Restricting the rebirth of Rome to Orthodox Greeks is, oddly enough, ahistorical.