Who is the real successor to Rome: The Byzantine Empire or the Islamic Caliphate?

I don't think it is worthwhile to debate whether one state was more of a successor to Rome or not, as both the empire and the caliphate drew upon their Roman past and both changed significantly during their existence. Moreover, contemporaries would not have seen it this way, most importantly because Rome did not fall in 476. What actually happened was that a weak regime ruling over what remained of the Western Roman Empire fell to an internal revolt led by Odoacer, who then transferred the imperial regalia (and thus authority) to Constantinople. There was no need for a successor, since there was now only one Roman Empire and everyone recognised that fact, with imperial titles and coins being common throughout the west well into the sixth century; unsurprisingly, the so-called 'barbarians' did not want to throw everything away, it was the world they grew up in after all. Moreover, the 'barbarian' kingdoms had very real claims to being successors to the Roman Empire as well. Their leaders were men who had served the imperial government before it was replaced, the people they ruled were still the same and Roman institutions did not disappear overnight, so I think it is absolutely fine to say that they were successors to the Roman state too. Concepts like empire or Romanness were very negotiable, so we shouldn't divide history into neat boxes either. New states will inevitably draw upon local history and cultural memory to enhance its legitimacy, and the 'barbarians' were no exception.

It gets even more complicated since the Eastern Roman Empire was doing pretty well in the fifth century (relatively speaking). Nothing changed for them in 476 and emperors would continue to rule the Roman Empire from Constantinople until 1204. The Fourth Crusade ended this, but sections of the old aristocracy took back Constantinople a few decades later and ruled the city until 1453, nearly a thousand years after Rome 'fell'. The Byzantine Empire was not a successor to the Roman Empire, it literally was the Roman Empire and 'Byzantine' is only a convenient term used by historians to distinguish between the medieval phase of the empire with the earlier period everyone knows about, so there really is no reason for anyone to argue that it was a completely different empire. Personally I'd rather use 'Roman' to describe the empire, but frankly it is easier to say 'Byzantine' a lot of the time and as long as everyone knows about what it means, it doesn't make that much of a difference. As I said above, institutions and cultures change, so even though the empire looked very different in 1204, 641 or even in 476 from the empire built by Augustus, we should still remember that it was still basically the same state.

More tangibly, the Romans had taken back Italy, North Africa and parts of Spain under Justinian I (527-565), most of which would remain under Roman control until the seventh century, with parts of Italy even until the eleventh. Throughout this period, Romans continued to interact with the West and both influenced each other; everyone in the Mediterranean world basically lived in the shadow of Rome, whether under the very real empire that was centred in the East or under the cultural legacy of the empire in the West, so in a sense everyone was a successor to Rome. Who are we to judge who was the 'real' successor? There are simply too many factors to consider and until we are provided with a very tight definition I don't think there is a right answer to this. I'll admit that my perspective is heavily influenced by my interest in cultural and religious continuity, and that others have different opinions, but that in itself illustrates how difficult it is to answer this question.

This is even before we talk about Islam. Islam was born into a world dominated by two world powers - the Roman Empire and the Persian Empire. The two imperial empires regularly intervened in Arabian affairs and Arab soldiers had served both states for centuries. Cross-cultural contact through trade, religion and political patronage made seventh-century Arabia a fascinating place to study and in my opinion Islam was a religion thoroughly rooted in the dynamic world of late antiquity, so Roman influence on the early movement was simply a given. One perspective on Islam that I agree with is that it began as an apocalyptic movement (which fits very well with the cultural milieu of the time) that explicitly included followers of various forms of Judaism and Christianity within the early Islamic community, which might explain why the conquests succeeded so spectacularly - there were after all plenty of Jews and Christians within the Roman Empire who might be attracted to this new force coming in from the south. Indeed, many of the Arab conquerors, once deconstructed, cannot be simple nomads from the desert, but soldiers who had served the two empires already but turned on them when the imperial powers were weak. They were in essence local auxiliaries who had gone rogue and united under the leadership of charismatic men preaching a new inclusive reform movement. They may not seem very Roman on the surface, but the reach of Rome was long and we cannot underestimate the role of the wider late-antique context in shaping Islam into what it was.

More obvious connections become visible when we look at the first decades of the caliphate. Local administrators were often kept in place, cities were taken by treaties more often than by force and different monotheistic beliefs were generally respected. If you were a Roman living in Syria who had a lot of dealings with the Arabs guarding the borders already, who visited churches built by wealthy Arab chiefs and who was perhaps even ethnically Arab yourself, how much did things really change for you? Very little I'd wager. The historian Robert Hoyland, who wrote an excellent and up-to-date book on the Arab Conquests last year, had this to say on this topic:

From this perspective the Arab conquests began as an Arab insurrection, that is, the early conquerors were not invaders coming from outside the empire but insiders trying to seize a share of the power and wealth of the Byzantine state. This helps explain why the Arab conquests were not particularly destructive: the leadership already had close acquaintance with the empires and they wanted to rule it themselves, not destroy it. [...]

But the Arabs did not only employ military means to further their aims. They also made heavy use of agreements to respect life, property, and customs in return for submission and tribute. Such agreements were part of an ancient Middle Eastern tradition of rules for military engagement, examples of which exist from as far back as the third millennium BC.

The Arabs were no different from other 'barbarians' trying to carve out a piece of the empire for themselves and thus have a perfectly good claim to being successors to Rome. Things got even more complicated when Arab rule was consolidated, since classical learning and Christian culture both continued to flourish within the caliphate. I can't really talk much beyond that since my knowledge pretty much ends at the seventh century, but the point I'm making here is hopefully quite clear: the caliphate was born in a world dominated by Rome and ended up ruling many former Roman provinces, so they definitely inherited aspects of the Roman past. As a historian I don't think I should pick which of the empire or the caliphate was more of a successor to Rome, as there are too many variables to consider and to do so would be picking sides, when both states are very interesting in themselves. The extent to which Roman culture changed under their rule is a fascinating topic and it shouldn't be reduced to a misleading comparison like this.