It's possible to argue that, sure, but if you really want to you can argue for a whole bunch of historical technicalities that don't make a whole lot of sense. I'm sure, for example, you wouldn't consider the English monarchy to be a continuation of the Gaelic High Kingship of Ireland, but the circumstance of the area being conquered and the title being claimed (in some capacity) is very comparable to that of the Ottoman conquest of the eastern Roman Empire.

I suppose it would come down to what you would consider a "continuation" in terms of historical politics, especially in the 15th century, where we don't have the notion of Westphalian statehood that we do today.

The Roman Empire as it stood in the 15th century was the same entity that had been founded by Augustus in the 1st century BC. It had certain cultural, political and social inheritances associated with its "Roman-ness", for want of a better term. The people who lived there called themselves Romans and I think that's as good a metric as any for deciding whether or not it was "really" still Rome at that stage. The Ottoman Empire was a completely different political entity founded under a completely different set of circumstances with totally different cultural, political and social trappings. Apart from a single Emperor of theirs who called himself "Caesar", the Ottomans did not call themselves Romans. To me personally that leaves the case very open and shut. Your question was whether or not you could make a case for it, and you definitely could, I just don't think it would hold very much water.

I think it would be fair to consider the various French Kingdoms and Republics to be continuations of each other, for example, as they represent different stately forms of a single French nation consisting of a single French people. The Roman Empire and the Ottoman Empire originated in completely different time periods in completely different parts of the world and were political representations of the interests of two entirely different groups of people. I don't really see how they could be considered the same entity. One conquered the other, but that happens all the time in history and people make much less noise about continuations and inheritance and so on. I wouldn't consider the Kingdom of Spain to be a "continuation" of the Aztec Empire, or the United Kingdom to be a "continuation" of the Ghana Empire, so much in the same way I wouldn't consider the Ottoman Empire to be a "continuation" of the Roman Empire.