''With the Conquest of Constantinople in 1453, Fatih Sultan Mehmed II (1432 - 1481) claimed the title Kaysar-i-Rûm "Emperor of Rome" and proclaimed himself the protector of the Orthodox Church. He appointed the Patriarch of Constantinople Gennadius Scholarius, whom he protected and whose status he elevated into leader of all the Eastern Orthodox Christians. As Emperor of Rome he laid claim to all Roman territories, which at the time before the Fall of Constantinople.
Sultan Mehmed II also took the title of Pâdişah, a Persian title meaning "Master of Kings" and ranking as "Emperor", claiming superiority among the other kings. His full style was Sultan Mehmed II Khan, Fatih Ghazi 'Abu'l Fath (Victorious Conqueror, Father of Conquest), Padishah, Sovereign of the House of Osman, Emperor of Rome, Grand Sultan of Anatolia and Rumelia, Khan of Khans of the Two Lands and the Two Seas, Emperor of the three Cities of Constantinople, Edirne and Bursa...
The claim was recognized by the Orthodox Patriarch of Constantinople, but not by Roman Catholic Western Europe. Gennadios (Georgios Scholarios), a staunch enemy of the West, had been enthroned Patriarch of Constantinople with all the ceremonial attributes of Byzantium by Mehmed himself acting as Roman Emperor and in turn Gennadios recognized Mehmed as successor to the throne. Mehmed's claim rested with the concept that Constantinople was the seat of the Roman Empire, after the transfer of its capital to Constantinople in 330 AD and the fall of the Western Roman Empire.
Mehmed also had a blood lineage to the Byzantine Imperial family
; his predecessor, Sultan Orhan I had married a Byzantine princess, and Mehmed may have claimed descent from John Tzelepes Komnenos. The Ottoman Empire also captured Otranto during that period, and Mehmed II was planning on taking Rome itself when the Italian campaign was cut short by his sudden death. The title fell into disuse after his death, but the imperial bodies created by Mehmed II lived on for centuries to come.
The Turkish historian İlber Ortaylı is a proponent of this claim, citing the multicultural make-up of the state and Sultan Mehmed's acceptance of certain Byzantine court customs. Professor Ortaylı finds Russia's claim to the title to be only nominal, and that Sultan Mehmed based his court policies and conquests on creating a third, Islamic Rome (the first Rome being polytheistic, the second one Christian).''
''When the Ottoman ruler Mehmet II captured Constantinople in 1453, this legitimized his assumption of the universalistic title Kayser-i Rum Emperor of the Roman Empire. It was more than window-dressing to appease the Christian majority among his subjects. The Roman notion of the world as empire a political unity under a single ruler and, since the 4th Century CE, a single god was a powerful political ideal that was deeply ingrained in popular belief and hadbecome associated with the possession of Constantinople, the Imperial City. Thus the capture of the ''Red Apple'' enabled the Ottoman dynasty to assume imperial pretensions, in defiance of the Mamluk rulers of Egypt and the Holy Roman Emperor in the West. This in turn allowed Mehmet's successors to appropriate also the equally universalistic titles of padishah (Great King) and khalifa (Caliph, ruler of Islam). To strengthen his claims to world dominion as well as, perhaps, for personal reasons Mehmet moreover chose Alexander the Great for a role model.''
No. You may consider the Byzantine Empire as a continuation of Rome, as the Roman emperor Constantine moved his capital to present day Turkey in the 4th century and built the city of Constantinople. The Roman Empire was eventually split in two for administrative purposes. The Eastern half survived the fall of Rome and became the Byzantine Empire. The Ottomans destroyed the Byzantine Empire in 1453 and created their own in its place.