The Caliphate arose as the ape of the Byzantine empire in the same sense that Islam arose as a universal religion to ape and replace Byzantine Christianity. The idea that an empire could be united politically through a universal religion that must ultimately replace and displace all other all deities in the world is, sadly, the result of the Constantinian synthesis.
Constantine saw in Christianity's monotheism and Christ's moral teachings a way to unite an otherwise disparate and far reaching empire that had been in steady decline for quite some time. The old paganism was seen as the reserve of a bloated and overbearing aristocracy, chiefly residing in Rome, and living off the fat of the rest of the world. Paganism was stale and antiquated, and it failed to inspire. Certainly no one would die for it though many would die because of it. In contrast, Christians not only were willing to die for their beliefs but they also lived virtuous lives. They also had, despite the periodic persecution that disrupted their lives, a network of accountability and a mutually recognizable order throughout the empire. This was not something to take lightly. Paganism had never developed the like. Some years later, Julian the Apostate, in his attempt to reinstate paganism, would mimic the Christian model of government by bishops, sees, dioceses and parishes, but to no avail.
But Christianity had an Achilles' heel that Constantine was confronted with almost from the start. Constantine embraced Christianity when it was not united doctrinally, when the Arian and Donatist controversies were well underway. Christianity proved to be a factional religion and once given status, imperial recognition and patronage, and, hence, an authoritative voice in society, it very soon turned against its own, condemning adherents of suspect positions. Even competing sees would be quick to seek political advantage in opposing each other's doctrinally-nuanced approaches to Christological positions.
The Christological controversies of the 4th-6th centuries proved in the end to be more divisive and more dangerous to the unity of the empire than the disintegration of the empire under a corrupt pagan oligarchy had ever been. Muhammad saw this. He knew just how to exploit this: by making the confession of his new religion the simple monotheism of the Jews, yet without the ethnic exclusivity; by embracing Christianity's universal outlook, yet without the doctrinal nuances that encumbered Chrisitan thought and caused schism. He could take Constantine's model and make it his own: the Caliphate.