(Here is some history you probably never knew. I quickly drafted this response to a Quora question, but thought it might be of more general interest).
The Roman Empire did not fall until 1453, when the Ottoman Turks conquered Constantinople. Emperor Constantine had moved the capitol to the Greek city of Byzantium because of its excellent geographic, strategic, and central location, calling it New Rome, although at that point the empire was then split between eastern and western halves until the west collapsed in the fifth century. Subsequently renamed Constantinople, it then became the center of the world and continued long after the city of Rome itself declined sharply as the locus of empire shifted. Attempts, sometimes successful, were made to reclaim parts of the west. Indeed the name of the city itself continued until the cataclysm of World War I.
In addition, Justinian was the last Emperor to use Latin rather than Greek, which prevailed for another eight centuries, or far longer than Rome itself. There was never a “Byzantine Empire.” That was a far later German concoction in order to legitimate their own “Holy Roman Empire, which, as Voltaire once said, was neither holy, nor Roman, nor an empire, What actually existed, as I have suggested elsewhere, was not something “Byzantine,” but rather a Greco-Roman Empire that lasted for a millennium. This is still confused and not well known due to the ignorance and prejudice of northwestern Europeans towards the south.
Thus, the Roman Empire did not fall or disappear the way you have been led to believe. In fact, if the empire began with Augustus, the Rome-based portion only lasted for a bit more than four centuries, while the Eastern Roman Empire continued for another thousand years, and all the people in the empire, including the Greeks, referred to and considered themselves Romans until the very end. The empire itself was polyglot as it always was, but the dominant language was Greek. Thus, while the “Middle Ages” or medieval period happened in the west, which was still backward, that was not the case in the east, in what in fact was the bulk of the empire, which continued a thriving, but heavily Christianized civilization. Thus, in the year 1000 Constantinople had a population of over a million people.
The original Roman Empire was far better at practical things, particularly organization, administration, functional engineering, etc. than the quarrelsome Greeks, who otherwise brilliantly developed and dominated culture, art, science, philosophy, etc. Romans provided the critical stability, legal and social infrastructure that was previously lacking in an otherwise largely Greek world that had been established by Alexander and his successors. As a result, Roman law, justice and organization provided the framework which sustained an otherwise largely Greek cultural entity.
The empire would not have fallen had it not been for the Fourth Crusade (1202–1204) which, instead of going to Jerusalem to fight Muslims attacked other Christians via Venetian ships and encouragement. The barbarian Knights/Crusaders literally destroyed the city of Constantinople, pillaging and worse, burning down much of the city while the Venetians, at least having some artistic sense, carted off a lot of treasures that you still see in St. Marks to this day. The crusaders took over what was left of the city setting up their own “empire” for a few decades until the Romans finally reclaimed it, but in a much-weakened state. Thus, by the time the Turks finally conquered the city in 1453 with new long-range cannons, there were only about 50,000 people left there. That is when the Roman Empire finally fell, in its Greco-Roman-Christian incarnation.
Given true history then, if the empire hadn’t fallen it is quite possible that Greek would still be the common language, you might be speaking it, and the European world would be very different. The “old” nations of Europe after all, have only existed for around five centuries. They really ascended after the fall of the City and the “renaissance” that followed, (another retroactively coined term), instilled to a considerable extent by Greek refugee scholars who fled to the Italian city-states, which welcomed then as teachers, given their thirst for knowledge and ancient manuscripts.
Many other things would be different, starting with religion. In the empire the church was subordinate to the Emperor, and less powerful than in the west, where given the power vacuum, the Bishop of Rome, or Pope, was able to claim supremacy. Consequently, if the empire had continued the church would not have had so much power, and it is likely there would never have been a Protestant Reformation. Regions would have had their own Patriarchs. There is no way to know how geography would have taken shape, given uncertainty as to how far the empire might have reached, or how competent it would have been, but at its maximum it could have stretched over all of Europe. Alternatively, it could have been smaller, as it continued to fend of endless barbarian invasions from the east, allowing other powers to emerge in the west anyway, but possibly without the virulent nationalism that eventually developed and led to catastrophic wars and mass murder in the last century.
Much would depend on whether the empire could regain Italy, which was essential to the emergence of modern Europe. (In 1500 Italy was richer than all of western Europe combined and more culturally advanced). Since it was then divided into competing city-states in the fashion of ancient Greece, that might have transpired if the empire was relatively free from barbarian invasions. Slavs, at least in the south were already being absorbed before the fall of the empire and thus would likely still be integrated. Unknowns are how the Russians, or more importantly the Germans might have emerged. But generally, the empire had always done a great job of absorbing and integrating other peoples, until the rise of Islam.
If the empire had remained strong it is unlikely that the Arabs would have conquered what was then European Christian north Africa and Egypt, or even much of what is now called the “middle east,” which would have remained Christian and Roman, linked to the west via Constantinople. The empire took the brunt of Muslim invasion, thus protecting the west for centuries, and if not for the Fourth Crusade could have grown stronger and more powerful and Europe would not have been threatened or invaded. In addition, Persian culture might still exist, possibly as yet another empire.
Greek rationalism did not disappear as the church in the east largely reconciled itself with it and focused on spiritual matters. Consequently, the church would not have been able to significantly thwart scientific progress.
Europe, including a good portion of the east would still dominate the world. and remain centered in Constantinople. They would likely still have found the Americas, but development might have been different, possibly without African slavery, among other things. The culture would be more advanced and sophisticated, but economic growth might not be as robust, given a different emphasis and outlook, and without a “Protestant ethic” but rather a sort of merchant mentality. Higher education might not have been found in universities as we know them, but rather closer to the academic format of the ancient world, more focused on the development of the mind and learning.
Ultimately we cannot know because of vicissitudes of one-man rule. There can be a great ruler and then a terrible ruler. You can have a Marcus Aurelius or a Caligula. So given different personalities there is no way to know, and that can be precarious and lead to instability or conflict until someone else consolidates power. What we can learn clearly from history is the less power we surrender to others over us the better off we are.
Finally, the Fourth Crusade was clearly the worst disaster in western history even beyond World War I if only because the latter might never have happened if the empire hadn’t fallen. Such a general and vague question can lead to endless possibilities and alternatives, but in the end all that is left is what actually happened.
