I'm not an expert on the Ottoman era but from the survey works I've read, Mehmet II did indeed initially consider his conquering of Constantinople to signify that the Ottomans had taken on the legacy of the Roman Empire. Just like Constantine had brought the Roman empire from Paganism into Christianity, he saw himself as bringing it to its next stage; from Christianity into Islam. I don't think the idea existed very long though, perhaps even before his rule ended he gave up the idea.
I'll see if I can dig up the work I read it in. It was a survey work, maybe Hodgson or Lapidus.
They made a lot of noise about being the successor of Rome, but the degree to which this was acknowledged varied a lot. A history professor on a lecture I was on said that the king of Poland addressed a letter to Mehmed II as "the emperor of Constantinople" and the pope offered to crown him Emperor of Rome if he'd convert to Christianity, which he turned down after seriously considering it. Catholic Europe on the eve of the Renaissance wasn't terribly fond of or impressed by the run down and (to them) heretical Byzantine Empire, so the initial reaction to the fall of Constantinople wasn't as strong as you'd think, but the initially ambivalent attitude turned sour in later years when the Ottomans started invading deeper into Europe.
Then again, the French were allied with them against Austria for a time, so the whole story isn't quite so simple.
/u/Vetmichael answering a similar question
Great question: actually the pretension to Roman authority predates the Ottoman conquest of Constantinople in 1453. As early as the Umayyad Caliphate, which relocated its capitol to Damascus, there were pretensions of Romanesque power and glory. (see: http://www.metmuseum.org/toah/hd/umay/hd_umay.htm )
Ottomans, such as Mehmed II, did adopt the airs of Roman authority - they had long pined to breach the walls of its most affluent and prestigious city, after all - but there is very little commonality between Ottoman institutions and the Roman (especially Western Roman) predecessors. For instance, the Ottomans eventually put the dying Byzantine empire out of its misery not primarily because they wanted to control the famed Roman city, but because it proved their right to rule and their divine guidance; several dynasties of Muslims had laid siege to Constantinople and failed. (see http://lostislamichistory.com/mehmed-ii-and-the-prophets-promise/ ). Of course, Mehmet II did not allow widespread killing and looting of the city, partly because of his respect for an age-old foe of Islam but also it didn't make economic or social sense; it was far more profitable to have the residents pay a jizya or religious tax, than it was to loot, pillage, and burn.
Ottoman imperial structure is more closely related to the Persian model, with strong influence from Arabacized Islam, and Turkish traditions thrown in; Roman affectations were probably just another feather in the cap of the Emperor. For example, the Ottomans relied upon the jannisaries or slave-soldiers to form the backbone of the army and the bureaucracy [something it did very well]. But there isn't really a slave army model in Roman history. There was no forum or senate, so the governmental structures of classically "Roman" society were not there; though the Ottomans did emulate the Byzantine emperors a bit, they were more closely emulating Caliphs by that time and not infidel emperors.
So, did Mehmet (and his successors) see himself as a "Roman" emperor? Not really, but there was a certain cache that went along with assuming the mantle, especially vis-a-vis Christian Europe. Interestingly enough, after the fall of Constantinople, the Russians endeavored to take up the mantle of Roman civilization, referring to Moscow as the "Third Rome" and the adoption of the title "Czar" to denote their claim of Roman heritage. [another side-note, the Russians aren't the first to use "Csar" or "Tsar" - the Bulgarians were using the title as early as the 10th century, leading one to assume that they also pretended to have Roman imperial blood/authority].