If not, what is the general consensus on when the Roman Empire really fell? Was it during the Heraclian dynasty in 7th century or maybe in 1453 when Constantinople was captured?
It would be a little silly, but sure. The Ottomans largely reformed the territory of the Eastern Roman Empire, maintained an effective tax based bureaucracy underpinned by an ethnically diverse civil service, and for a time at least they styled themselves as the Emperors of Rome.
I think it would be more interesting, however, to say that the Ottomans represented the end of a particular Mediterranean process. The Ottomans were really the last great Mediterranean empire, a form of state that had its apogee with Rome, and its demise represents the final passing of that political form in favor of the nation state, for good or ill.
If you want to be a bit silly... you could say the kingdom of Spain is currently the Roman Empire, because the heir of the last Byzantine emperor sold his titles and claims to Ferdinand and Isabella.
Whereas the Byzantines were very much Roman other than the language they spoke, the Ottomans were not. They were different both culturally and ethnically. I would argue against it. Especially considering if you follow that line of logic, Turkey remains the last bastion of the Roman Empire since it is the successor state to the Ottoman Empire and holds Istanbul/Constantinople as its capital.
I've argued it before, but the true last bastions of the Roman Empire are both the Eastern Orthodox Church, and the Catholic Church. Their power structures have more or less remained intact since the days of the Roman Empire, but considering neither of them are Empires, that's a hard point to argue.
I would have to resolutely say no, I would not argue that the dissolution of the Ottoman Empire in 1922 was or should be considered the final "Fall of the Roman Empire", and I think that it would be very hard to make a valid argument as such. The reasons for why I believe this to be so are relatively complex, but I will try to summarize my points in a clear manner below. I will also try to address some misconceptions that have popped up in the thread.
First, we must establish that the Eastern Roman Empire/Byzantine Empire was the continuation of the Roman Empire after the fall of the West. It was not a "successor", it was a legitimate remnant of the Empire, with its capital centered at the "New Rome", Constantinople. This new capital was established by a legitimate Emperor (Constantine I), who held control of the entirety of the old Roman Empire (politically, the same Empire that was ruled by an unbroken line of Emperors which included the great Augustus and Marcus Aurelius), and he ruled over a population that considered itself, without any doubt, overwhelmingly Roman.
The Christianization of the Empire, long a point of contention, had become tolerated by the Edict of Milan in AD 313, and by the time of Theodosius I at the end of the 4th Century, Christianity had been declared the official and sole religion of the Empire. Since Theodosius was the last Emperor to rule over a united Roman Empire (and also the one who officially split it), his establishment of Christianity as the official Roman religion is also legitimate, especially considering that, by this time, Christianity had become supported by a vast majority of the Empire's inhabitants, and to some extent, became a defining feature of what it meant to be "Roman" in Late Antiquity.
Throughout the 5th and 6th Centuries AD, the Eastern Roman Empire remained the dominant power in the Mediterranean. People living within the Empire still considered themselves Roman, and even invaders who had established new kingdoms in old Roman territory in the West still acknowledged the Eastern Emperor as the legitimate leader of the Romans, and in some cases, of the known world. In religious matters, the Emperor, the Patriarchy, and the Eastern bishops remained an overwhelming force in religious councils, despite a bit of (mostly futile) contention from the Pope. This meant a lot considering that Christianity now encompassed not only people living within the Empire, but also many of the "barbarians" living outside of Imperial borders. In essence, the religious decisions of the various Church Councils affected pretty much everyone under Christ.
Under Justinian's Renovatio Imperii, the Empire looked like it was going to restore itself to its old boundaries and glory. But the Plague and a series of devastating droughts probably caused by the strange "slanted light" of the late 530s, ensured that this would not be realized. Still reeling from the loss of perhaps a third to a half of the population in the mid 6th Century, the Empire began to observe serious issues in raising the large tax revenue and feeding its armies and population centers. In the end, this undermined the gains made by Justinian, with the overextended conquests in Italy and Spain weakening the defensibility of the Empire against foreign invaders. With a number of usurpations and revolts in the late 6th and early 7th Centuries, the days of the overwhelmingly powerful Roman Empire became numbered.
At the tail end of this disastrous period followed the final and apocalyptic Sassanid War under Heraclius, and then the world-changing Arab Invasions - the first, which critically weakened the Empire, the second, which almost destroyed it and caused the loss of over half of its territory in the span of a few decades. The middle of the 7th Century is where some people demarcate the end of the Roman Empire, simply because the Eastern Roman Empire no longer had the strength or authority to control a majority of the old territory it once held, and became confined to the lands of Greece, Macedonia, Thrace, Southern Italy, and Asia Minor.
From this point onwards, many scholars (myself included) refer to the Empire not as the "Roman Empire", or "Eastern Roman Empire", but the "Byzantine Empire" (a medieval Empire, similar in name and heritage, but different in scope and mindset than the classical Roman Empire). This is partly because the Empire had lost so much of its military power and administrative authority by this point, but also because Constantinople became the Empire's sole focus. It not only remained the best defended city in Europe, but was also built up to be the most populous, most cultured, and the most grand (hence the often used title "Queen of Cities"). All roads led to Byzantium. While it had been very important in centuries prior, after the 7th Century, it became, for all intents and purposes, the heart, mind, and soul of that medieval Empire.
But this raises the important question: was this diminished Empire (post-7th Century) still the Roman Empire? Well, in the process of trying to answer that question, let us list a few important observations about the medieval Byzantines and their important links to the "old Roman Empire".
Byzantium retained an unbroken line of Emperors stretching back through the old Roman Empire to the time of Augustus. While not every succession was pretty, each Emperor succeeded the next according to the customs and features of historical Roman political succession. There were no significant gaps in leadership, and most, if not all rulers came from locations that were considered a part of the Roman world.
Byzantium also retained a majority populace that overwhelmingly considered themselves to be Romaioi "Romans". While the language of the administration may have changed (and this is not really as major a change as you might think - Greek had always been the second language of the old Empire), the people felt they shared a common heritage with the old Romans, and that the Empire was simply in a temporary lapse. They believed that one day the lands that had been lost in the West and South would be liberated, absorbed back into the Roman world, and that the Empire would ultimately endure.
Byzantium remained a staunchly Christian society, despite various schisms and heresies, and the state religion remained a defining feature of Byzantine citizens, as in the Late Roman world. As the centuries drove on, it became more and more cemented in Byzantine traditions and society, even if not everyone truly believed in it. Byzantine Greeks called themselves not Graikoi "Greeks", but Romaioi "Romans", because the former had the connotation of being a pagan, while the latter came to signify a Greek of the Christian faith.
Byzantium continued to be a culture very much defined by its military, as with the old Roman Empire. Since the military was both a vital asset to the very survival of Byzantium, as well as a strong cultural force, it was naturally idealized by the citizenry, who venerated the military saints and glorified military leaders. To further this point, we note that the strategos was often times both the military leader of a theme, as well as the political governor, a clear indicator of the ties between governance, society, and the military in the Byzantine world. The adaptability and versatility of the Byzantine military, the continued use of Late Roman-styled units and names of units (for example, the Klibanaphoroi and Scholai), as well as the use of old Roman vexilla in warfare also emphasizes this veneration of the old Roman military tradition.
If we jump forward and compare these traits of "Romanness" inherent in medieval Byzantium to those of the Ottomans, we gain a number of interesting findings and conclusions:
First and foremost, the Ottomans were not a part of the great unbroken line of Roman and Byzantine Emperors, nor were they part of the group of peoples that were considered "Roman" at the time. They were peoples that had historically been great enemies of the Empire (and who generally refused to be considered Romans themselves), and were one of the groups that contributed heavily to the Empire's ruin by conquest. The Ottoman predecessors, the Seljuqs, had tried to emulate and usurp Byzantine power in Anatolia by naming their state the Sultanate of Rûm, a name taken to legitimize their claim in those lands. And unlike the Romans or the Byzantines, the indigenous peoples conquered by the Turks were not kept in place, but were subverted or removed entirely, as noted by events such as the disappearance of the Romaioi in Anatolia following the Battle of Manzikert in AD 1071. Likewise, Mehmed II followed this trend by taking the title of Kaisar i Rûm (Caesar of Rome) in order to legitimize his taking of the Byzantine capital, not because it was bestowed upon him by anyone with the authority to do so. His claim to be descended from John Tzelepes Komnenos would not have made him a claimant to the throne, since not only was the Komnenos family no longer in power (there was a family of Grand Komnenoi that fled to Trebizond and ruled there after the Sack of Constantinople in AD 1204, but their rule was considered by the restored Byzantine government in Constantinople to be little more than a regional rebellion), but this certain John Komnenos that had renounced his religion (twice!) and fled to Muslim lands never to return, would have been branded an unpardonable traitor under Byzantine law. Therefore, there is almost no legitimacy to Mehmed's claim.
The Ottomans, while being somewhat tolerant of Christianity, did not adhere to Christianity themselves, which is also a major part of why they cannot be considered claimants to the Roman/Byzantine title. As we talked about before, being a Christian was an important feature of the Roman/Byzantine citizenry from Late Antiquity all the way until the Fall of Constantinople in 1453. At no point in the Byzantines' thousand-year history did they accept Islam as a religion, and contrarily, they fought for those thousand years for the Empire's very survival against the adherents of the Islam. The Muslim faith was always disliked by the Byzantines (to some, it was just another heresy - a very dangerous heresy that robbed the Empire of about half of its territory), and at no time did Byzantine viewpoint on Islam suddenly change.
Despite Mehmed's lofty rhetoric regarding his restoration of the Roman Empire, his initial claims did not seem to last long. Few, if any of his successors pressed the idea that the Ottoman Empire was a restored Roman Empire, even when they wielded supreme power over the Mediterranean world. Even Mehmed himself seems to have abandoned his claim soon after he conquered the city. This furthers my point that Mehmed's title of Kaisar i Rûm and his restoring of the Patriarchy and his tolerant views on religion were merely initial attempts at gaining the support of the remaining Greek population or wooing doubtful Westerners as an "Imperial shoe-in". Since many Byzantines fled to the Italian maritime states during and following the collapse of the Empire, it tells us that these people distrusted Mehmed and did not support his claims as Roman Emperor. His swift transformation of the Hagia Sophia into a mosque betrays an underlying reason for conquering Byzantium - it was not about really about restoring the Roman Empire as he tried to lead people to believe, it was about gaining power for the Ottoman state, it was about fulfilling the hadith "Verily you shall conquer Constantinople. What a wonderful leader that leader will be, and what a wonderful army will that army be!", bringing great glory and prestige to himself, and finally ridding the Islamic world of the annoying presence of an ancient and once powerful enemy that had fallen from grace. Finally established at the once great capital of Christendom, Mehmed had the authority he needed to claim the title of Caliph, a title that had been long elusive in the Islamic world after the decline of the Fatimids and the Sack of Abbasid Baghdad in AD 1258. The powerful influence of the Ottoman state because of the restoration of this title is undeniable.
In regards to the military tradition of Rome, I think I can easily sum up the point by posing the question: to what extent did the Ottomans preserve or venerate the long history and tradition of the Roman or Byzantine military? I believe the answer is: they didn't. Not only was the Byzantine military too weak by the 13th Century to have left a lasting impression of their military on the Ottomans, but the Turks had a long tradition of successful tactics in war against Byzantium which they would have been very hard-pressed to give up. In essence, there was very little that the Roman military tradition could have provided to Ottoman warfare.
So, in conclusion, we find that the Ottoman claim to the title of "Roman Empire" is tenuous at best. It would seem that the idea that they were successors and supporters of a Roman Empire was merely an attempt to legitimize their actions against Byzantium during their rise to power, and continued in later periods only when it benefited Ottoman rule. Through the centuries, Ottoman leaders did not consider themselves Roman Emperors, and their subjects (besides perhaps Ottoman Greeks) did not consider themselves to be Romans. In this regard, I believe that the evidence speaks for itself - the dissolution of the Ottomans in 1922 cannot be considered the final Fall of Rome.
After the Crisis of the Third Century and the ensuing division of the empire into east and west both Rome and Constantinople were capitols. Thus when the western empire fell the eastern or Byzantine empire remained. A lot of people do not seem to get that the Byzantines never called themselves Byzantines they called themselves Roman in various Latin then Greek form. In 1453 when that empire fell to the Turks under Mehmet II he proclaimed himself as a Ceasar of "Rum" or Rome. So in a sense one could argue that the Ottomans were the new Romans and when they fell in 1922 that would have been the end of the empire.
This is more a semantic argument, that I myself frequently indulge in, for the purposes of demonstrating the difficulties revolving around questions of Roman continuity and definitions of both "Roman" and "continuity."
Logically one can certainly argue on the basis of particular definitions of Roman imperial continuity, namely with titles (Ottoman sultans as holders of the title of Caesar of Rome as conquerors of Constantinople) and lineage (the Ottoman family having blood descent from Byzantine/Roman emperor Isaac Komnenos).
However legitimacy, is different because it is a social framing question, as in "which groups of people accept this argument as plausible."
If your social bounds for the legitimacy of Roman continuity into 1922 is hyper-Romanist Byzantine-Ottoman imperial apologists, then yes, certainly by this definition, the aforementioned argument is plausible. Because they would have a political purpose in seeking to portray such an extensive continuity.
However, if your social bounds for legitimacy is the overall popular world, as in "your day to day average joe who's heard of the Roman empire and sort of heard of the Ottoman empire", or the academic world (who have very specific and frequently differing ideas as to what counts for continuity) the answer to legitimacy will likely be "no."
tl;dr - Sure you could argue. The question is who would accept such an argument.
The "Byzantine" Empire was always the Basileia ton Romaion, and its inhabitants called themselves Romaioi. The primary identity was Roman. The Ottoman Empire was the Devlet-i Aliyye-i Osmâniyye, not Rum. If you were to ask a Turkish inhabitant of the empire what he was, he would say that he was a Muslim and a Turk. A Greek inhabitant perhaps would call himself a Roman, but he would also say the Sultan is not Roman. The primary identity of the Ottoman Turks was not Roman. A sultan calling himself the Caesar of Rome is just another monarchical title without much meaning, much as the King of Spain still has the title of King of Jerusalem.
It seems to me that the Ottoman rulers took the title of Ceasar of Rome as a right of conquest, but they never called their empire Roman. Similarly, British monarchs were Emperor of India but the British empire didn't become an Indian empire.
Can you call yourself a Roman if Rome is no longer part of your empire?
It may be farfetched, but i'll reword it through two other questions. Was the Mongolian Empire just an extension of Chinese dynasties? If Britain had collapsed after the American Revolution, would the United States' existence today be an extension of the British Empire?
To make this argument stick, I think you would have to prop it up on some theory that argues empires in general are best thought of as continuous given some condition that happens to remain constant across the Roman and Ottoman empires. And this theory should do a lot of useful work to justify identifying empires in this way rather than by other means.
Who would today's "heir" to the Romans then?
Depends on who you ask. You ask a guy living in Asia Minor who they were at any given point in time they might call themselves Roman or Ottoman or Turkish or Arab or any number of things. Are people from Germany Germans or Romans?
There isn't any consensus on when the Roman Empire fell, because it didn't fall, in the sense of being conquered by a foreign power or coalition of foreign powers as a result of a particular military campaign at a particular time. The Roman Empire fell apart, gradually, over the course of centuries.
Look at, for example, the loss of Roman control of Brittania: nobody took it from them. They withdrew the troops that were on the imperial payroll in order to station them elsewhere, leaving the locals to defend themselves against any external threats, and then they stopped bothering to appoint governors, and eventually the area was no longer part of the Empire.
The east/west split is another example. That didn't happen because of foreign invaders. It happened for internal political reasons.
These same sorts of things kept happening, until eventually distinct nations with non-Roman identities emerged throughout the former empire's territory.
There are some cases where portions of what had been the empire were conquered by foreign powers from outside the former empire. The fall of Byzantium is a good example of this. The migration of Germanic peoples into what is now England could be another example. But there are also numerous examples where a nation that used to be part of the empire was invaded by a now-foreign power that also used to be part of the empire. Because the empire was no longer an empire. It fell apart.