Interesting concept: imagine the south had seceded successfully in the 1860s, then in turn had been overrun by an alliance of Mexico and European superpowers, and Canada had taken New York, New England, and ultimately the eastern boundary of the U.S. is the Mississippi River and there’s a new capitol built from the ground up in Colorado.
Would that still be considered the same nation as the one that won the Revolutionary War even though none of the territory is the same as the confederation from 1776?
When the Seljuk Turks conquered most of Anatolia from the Byzantines in the 11th century, they called themselves the Sultanate of Rûm (i.e., the Sultanate of Rome). The Seljuks were in turn defeated, first by the Mongols then by the Ottomans. The latter went on to conquer the rest of the Byzantine Empire. After the fall of Constantinople, the Ottoman Sultan Mehmed moved his capital there and styled himself “Kaiser-i-Rum”.
I think this is a bit of a hangup. A reasonable and BIG issue, mind you - being an empire and all ;). But a hangup.
The Byzantine state was momentarily in bad shape when Charlemagne had himself crowned and reinvented the western imperial title. But otherwise throughout most of the early Middle Ages the Byzantine state was easily the wealthiest and most militarily powerful state in Christendom. It’s military dominance slipped after Manzikert, but its superior wealth continued right down through the end of the Komnenoi era in the late 12th century ( contemporaries were in awe of the riches of Manuel Komnenos ). It was preeminent in prestige even to the western Europeans. It was really only after the 4th Crusade that it went into a semi-permanent tailspin.
If you posit a strong ruler after Constantine VIII ( perhaps a son of Holy Roman Emperor Otto III and his daughter Zoe 
), I at least can posit the Byzantines handily weathering Manzikert, pre-empting the Crusades and taking back the Levant after Malik Shah’s death. Egypt? Italy? Maybe.
And maybe that is too Crusader Kings II :). But I think you are overplaying the ‘universal imperium’ ideal as the sine qua non of Roman-ness. I think the Roman empire in 500 under Anastasius I was still the Roman empire even after being shorn of western Europe.
That’s what I had in mind.
Pretty sure they weren’t “the people in Constantinople” - which is what I said.
You have to keep in mind that the city of Constantinople had great meaning. It had stood for over a thousand years and withstood a number of previous attempts to besiege it. It was the heart of the Empire.
The feeling in Constantinople in 1453 was generally that the current situation was bad. But hey, times have been bad before. Remember when the Goths attacked in 378? And remember that time when the Bulgarians attacked in 580? Or when the Arabs attacked in 674 - that siege lasted for three years. Or remember how bad things were after we lost at Manzikert in 1071? Bad times. Or that time when the Franks took over in 1204? It took fifty years to get rid of those guys. Or remember the plague? Or the great fire? Or the Mongols? And these Turks think they’re going to take over and rename the country? Picture that happening.
My point is that there had been several periods in Constantinople’s long history when it looked like the end. And every time, the Empire had survived and been restored.
So when Constantinople actually did fall (turns out gunpowder was the secret) it shocked not only the people in the city but all of Europe as well. It was regarded literally as one of the biggest events of the millennium.
It was wealthy and contained awesome artifacts and buildings, like the Hagia Sophia, that awed contemporaries. Not to mention the sheer hugeness of the walls of Constantinople.
But what it wasn’t, was the ‘heart of a universal civilization’. Contemporaries were awed, sure - just as they were by (say) the wonders of Egypt. What they wanted, though, was Byzantine cash and access to its heritage of “classical” learning. They did not admire its current state, or wish to emulate it - save for those in Kievian Rus who admired access to Orthodox Christianity (and who might have, had the cards fallen otherwise, adopted Islam instead).
To my mind, that’s what we remember the Roman Empire for - that, in its time, it appeared to itself and to its neighbours as both the inheritors and current exemplars of civilization itself (ignoring for this purpose those inconvenient Persians 
).
It seems oddy inconsistent to argue about what it means to be “Roman” by claiming that physical possession of the “eternal city” of Rome itself is not significant, but on the other hand that physical possession of Constantinople is all-important.
My point is rather a simpler one: that long before Constantinople had in fact fallen to the Turks, the idea of what Rome represented - the beating heart of a unified civilization itself - had dissipated, into a bunch of kingdoms - Germanic, Slavic, Greek, and (later) Islamic.
As it turned out, despite the best efforts of Hapsburgs and Turks, it was never to be revived: there would be no more unitary “center of civilization” after Rome (ignoring, of course, Persians, Indians, Chinese, etc.).
We already took over, but on the quiet so as to not disturb your productivity. Don’t tell anyone about this.
You know, the standard history of Rome will note the many Greek influences on the empire, but the name probably does give short shrift to its Greekness. In fact, I’d say the empire lost much of its classic Latin “Roman” character around the time any Thracian strongman could become emperor.
Rome had been declining in importance to the Empire well before the loss of the Western provinces - Italy itself lost special status under Diocletian in the third century, becoming just another province (for that matter, as early as the 1st century BC, the Eastern portions of the Empire were recognized as the center of wealth and power - which is why Anthony (the senior of the Triumvirs) assigned Augustus rule over Italy, while taking the eastern portion of the Empire for himself). Constantinople was regarded differently - it was always treated as an imperial city, partially because its strategic location made it much more important to the Empire, and partially because it had been designed to be the home of an Emperor.
While poor Lepidus got the shaft from the get go - “Well, you get… I unno, North Africa or something ? But not including Egypt. Only the shit part.” 
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The Byzantine state was momentarily in bad shape when Charlemagne had himself crowned and reinvented the western imperial title.
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What tickles me is that, prior to that point the Byzantine Emperors kept tabs on the happenings in the Western kingdoms via the Church and although they had absolutely no administrative presence (or real jurisdiction) there would dole out official titles to this or that local strongman in the name of Rome.
Oh, to have been a fly on that wall… I picture some hairy-ass brawler holding court in a big hut amidst beer-pounding beefy Teutonic bros, getting mail they couldn’t read passed along by the clergy and receiving token golden doodads from some admirative stalker he hasn’t even heard of. “I bring news, Chief Bloodaxe ! You’re a proconsul !” 'Told you you shouldn’t have tapped that Wisigoth skank, Erik ! It’s not contagious, is it ?"
nm
I suppose the justification was that it had little to nothing to do with the Romans that founded the Empire. For the most part, Italy wasn’t part of it, they stopped speaking Latin, it included different ethnic groups and even defined themselves in opposition of the Latins (who in most wayshave a better claim to being descended from the Romans). In a lot of ways it was Rome in name only.