Would Europe ever have a Renaissance without a large population of Byzantine's fleeing to Italy? Would Europeans have discovered the Americas without the incentive to find alternate routes to trade? Overall, would, “Western Civilization” be in a better, worse, or just different place?
This is a more interesting question than it initially lets on. What would happen if the Byzantines, the last and legal vestige of old Rome, didn't completely fall? Questions questions.
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Well, first off the Ottoman Sultan would not have had any legal backing for his title of Sultan of Rum (Rome), as the old Roman Emperor is still technically kicking. This gets into wave after wave of barely-substantiated claims (by 800 AD it's arguable that the Byzantines could even truly be called Romans anymore; they continued the legacy, yes, but they were becoming increasingly Greek even then; does a single city-state have any more right to the title of Emperor of Rome than a Muslim Turk?), but technicalities considered the Ottoman Sultan would be a bit more shamefaced at somehow never coming to totally posses the namesake that the Ottoman Empire began with, the Sultanate of Rum.
The more interesting question is what the political landscape of this city-state looks like. You can't be an Empire with only a city and the barest bit of surrounding countryside. A city cannot function like an Empire. But what can a city function like? Well, Greek, Roman and Italian history tells us: a city can function like a Republic.
There is no way the Turks couldn't have taken Constantinople, given enough time. The only way they would have survived was if the Turks decided to allow them. The scenario, as I imagine it, is thus: the Turks want to keep their Greek citizens (they own most of Greece by now) pacified and want to make a show of allowing Orthodoxy to coexist with their realm, so the Sultan decides to officially “return” the title of Sultan of Rome to the Byzantine Emperor, who (grudgingly) must accept the shameful ceremony to protect his control over Constantine's city. The Sultan then declares his deference to the “most ancient Emperor” of Rome and declares that the entirety of the Ottoman Empire is at the service of Rome, to protect and defend the glorious Empire. It's a sham, of course; the Byzantine Empire (or as we should call it at this point, the City of Constantinople) has become a protectorate under the Ottomans. It is safe, yes, and nominally independent, but the Ottoman Sultan has managed to shame the Emperor far more deeply than failure ever could.
Surely simply keeping the Greeks happy isn't enough to let a poorly-defended Metropolis like Constantinople remain unscathed, though. But what could the Ottomans really want the city of Constantinople to be “independent” for? Well, that's relatively easy; trade. The rest of the states of Europe aren't going to be too keen to trade with Muslim invaders, but could they ever refuse to trade with the Emperor of Rome, a fellow Christian? No, of course not. And the Ottoman Emperor doesn't need to let those states know (though they'd suspect) that he skims the tariffs and collects the taxes from any trade going through Constantinople. And the Byzantine Emperor, shamed and useless as he has become, could never admit it and hope to keep the throne.
If this were to happen, what would the Byzantine Empire become? I'd hazard to guess it would become like an Imperial Republic, with the Emperor gradually diffusing his power down between magistrates and even eventually elected officials that oversee the Empire's trade, in a way coming full circle back to what Rome began as, albeit with a limited-power Monarch at the top (think modern Constitutional Monarchies).
Initially the Sultan would need to fund its naval buildup, but Byzantium eventually becomes the Ottoman Empire's trading department. The Empire has minimalist trade, only going to the Islamic world, whereas the Byzantines trade with the West on informal behalf of the Sultan. Their fleet would not rival Venice's; it would absolutely crush it. Only the British fleet of the 1700s could really compare in terms of size, as the merchant marine of Constantinople must trade with most all of Europe and haul their cargo back throughout the vast network of Ottoman cities, not to mention the galleys and men-'o-war they would require to defend their shipping lanes from pirates.
The initial strategic value of this decision cannot be understated. A historically low point in trade, when the Ottoman state was only making money by trading within its own borders and to other Islamic states, has become a booming period. Turkish citizens can now grow/make goods and sell them to Constantinople to trade off to the agriculturally starved West, in turn granting the Ottoman state taxes on its citizens' produce, on the exchange into Constantinople, on the tariffs for the trade, and on the final trade's sale via the Emperor's taxes on the ship's return; a four-way profit that not only grants the Sultan constant income across the whole year, but also enriches his citizens. A wealthy Empire would not need more conquest as the trade-starved Empire of history did, but I see no reason to assume the Sultans of this new timeline wouldn't try anyway, and they may well be more successful; higher demand from European markets means more men staying home to work the fields, yes, but the Ottoman army is well-armed, well-supplied, and can afford to buy mercenaries, slaves, or even fund military schools to staff its ranks. Malta and Sicily may very well fall to the Sultan, as might Hungary.
In the long-term, though, the Byzantine Republic is going to cause problems for the Sultan. Being his only major source of European trade puts them in a unique position to demand things from the Ottoman Emperor. Long reliance means the Ottoman merchant marine is weak and short-range, meant only to trade with Egypt, Arabia, and the Barbary States. Its anti-pirate fleet is much smaller than the Byzantine navy, and very poorly-trained compared to historical standards; any war the Ottoman Emperor is in the Byzantines will be forced to join, and their navies can easily behave as the Ottoman navies in the field.
Representing the Sultan's trade capacity, you can be sure the Byzantine Republic could demand (and probably get) some territorial cessions to assist in their trade. Crete, Malta, a few of the Ionian islands and perhaps Tunis are all good bets for islands/cities the Byzantines would request to expand their trade reach, and the Ottoman Sultan would be likely to agree; the further his merchant marine can reach, the more money the state can make.
In the long-term usage brings reliance, and the Ottoman Empire is at the mercy of the Byzantine Republic's trade capacity. The Byzantine Republic will start to gain steam. The vast wealth flowing into Constantinople, damaged by the Sultan's skimming but not destroyed, brings riches to the Emperor. The new ports granted by the Sultan grant access to lands to smuggle the Emperor off to in the event of war, and naval docks for their ships to shelter in between raids on Ottoman territory. Their control of the Orthodox Patriarch in the safety of Constantinople ensures the loyalty and alliance of Russia, a long-standing Byzantine supporter, and the Emperor himself can call the Greeks to arms.
What would eventually happen in such a case is anyone's guess. The hypothetical Byzantine Republic would not be powerful, not in terms of men, but its navy would be great, its nominal authority over the Greeks could punch the Ottomans in a historically unstable province, and the alliance with Russia could cause more than a few problems for the Turks. Would the Byzantines reclaim their former glory? It's doubtful, but it is possible that a well-timed rebellion and a strong naval blockade could cut the European portion of the Ottoman Empire off from the Asian, and potentially force the Sultan to cede the European side to the Byzantines. It would take a military genius and it's a long shot even then, but a pseudo-Imperial Republic called Rome could be reborn from the ashes of Constantine's city.