What if Constantinople and Ctesiphon Had Held the Arabs in Check?

C. 600 AD two great Empires controlled all the land from the Nile to the Tigris and beyond. These two frienemies had faced each other for centuries. Suddenly, they decided to have their worst and biggest war ever. In the end, it didn't matter who won, they both would be doomed by a new religion being birthed in the Arabian Desert. BUT, what if they hadn't spend the better part of two decades trying to annihilate each other immediately prior to the emergence of the Arab Muslims? What if the Arabs had come up against these two super-states and lost?

The Middle East would have been split between Zoroastrian Persia, Christian Rome (what we call the Byzantine Empire), and the Muslim Arabs. What would have been the eventual outcome? Would Islam finally have made rapprochement with one or the other in order to conquer the other one? Would Rome (Byzantium) and Persia have gone back to fighting each other after their brush with fate at the hands of the Muslim armies? Would Constantinople and Ctesiphon have made an alliance to defeat the new threat before returning to their continual fighting? Would the region's three-way religious split have continued to this day?