I'd like to provide a similar perspective as what I just described in my prior blogging about Jesus and his selection of Apostles about why God chose the Prophet Muhammad (PBUH). This is mostly to help Christians understand the context of Islam better by comparing the circumstances that the Apostles found themselves in after Jesus with those who needed to succeed the Prophet (PBUH).
Just as Jesus had a public ministry of only about two years, the Prophet Muhammad (PBUH) had only ten years over this new born Islamic faith before he died. While Jesus had a well established (too well established, as sin had become institutionalized) Jewish faith base to instruct and convert, the Prophet Muhammad (PBUH) was reaching back further in time in order to restore awareness and faith in the God of Abraham, back to Adam, and fill in a huge gap in faith history for what would be the Islamic people. God therefore chose a man who could be extremely strong and "a quick learner" (to use a vernacular term) because the Prophet (PBUH) had to receive a "crash course" in religious doctrine from the earliest times of Adam, through Abraham, and then the branch of descendants that led to the people of Islam. So while Moses was a "broker" of bringing the Law of God to man, and Jesus brought fulfillment of the Messiah who would bring the New Covenant for both Jew and gentile, the Prophet Muhammad (PBUH) had to be almost a religious archaeologist, learning from God through the angel Gabriel their lineage from Adam to the present time. This required a unique, strong, intelligent and forceful man of great faith and obedience. The Prophet Muhammad (PBUH) literally had to rediscover the locations of the faith of the Abrahamic family (shared by Jew, Christian and Muslim as the common tree) but also articulate how their branch of the Abrahamic faith came into being in their environmental and cultural milieu.
The Prophet (PBUH) died of natural causes in 632 after an illness, so his death was not a surprise or sudden event. The Prophet (PBUH) and his people were literally surrounded by violent tribes who were pagan and would never accept the Abrahamic faith of Islam. Islam could have been wiped out in theory before it even started if they did not, as the Israelites did before them, have a very aggressive stance against pagans and other unbelievers. This is one reason that Islam is very robust in its stance, as it literally would not have survived one generation otherwise.
Further, the Prophet (PBUH) left through his diligence not only the Qur'an but a wealth of his sayings, teachings, and life examples so that his successors could cope with his loss, which created an enormous vacuum. Remember he did not have the advantage of a priesthood, as did Moses, or the Apostles and disciples, as did Jesus, to perform as his successors and to nurture the new born faith structure.
In this context the uncle of the Prophet (PBUH) and others of his family and circle asked the dying Prophet what they should do, and who should succeed him. All sources agree that while the Prophet (PBUH) heard and understood the questions he remained silent. The Prophet (PBUH) died without appointing anyone to succeed him. Remember, this would be quite alarming and shocking in a culture that is structured to its core by blood, tribal, social status and, with the introduction of Islam, "the length of the period of companionship" with the Prophet (PBUH). Just as Jesus being convicted and killed was a scandal, the Prophet (PBUH) dying without naming a formal successor would have been quite shocking to his companions. But clearly the Prophet (PBUH) knew that God's will would be that the faith would have to live in the structure that would flower after him and as their first lesson in trusting in God after the Prophet had departed in death (PBUH). Again, this shows a great strength of character in a man with these social and cultural circumstances not laying his hand upon a successor when obviously he would have had strong personal opinions. He wanted his companions to live the faith with trust in God from the very moment of his death.
Much bloody war had to take place upon the death of the Prophet (PBUH) to keep the faith of Islam intact against both hostile pagan tribes around them and also what Christians would call "heretics." Just as Christians had people second guessing Jesus after he died so too did Muslims have rival claimants to Muhammad's prophetic authority. It's beyond the scope of this particular blogging to trace how this resulted in confusion about succession and authority that linger to today, flaring up into conflict in some circumstances. The point I wanted to make is that people today tend to think that Islam was the "bully on the block" from the very beginning and that is not true. Just like the Israelites several thousand years before them for the Muslims it was a matter of fight or be destroyed by surrounding hostile forces. The Christians had the advantage of using as their springboard the conversion of established Jewish faith from the Old Covenant to the New Covenant, so unlike the Israelites of Moses and the Muslims of Muhammad (PBUH) the Christians were very much individual-by-individual, and not a body of people affiliated by tribe that had to cut a swath to survive in pagan surroundings. Christianity did face its own internal convulsions, though, through violent actions of heretical factions.
I hope that this is illuminating and helpful to understanding why a particular man is chosen, and how God guides a man in the context of where he and his companions are in religious history. One of my favorite mental images of the Prophet Muhammad (PBUH) is exactly this role of his in being almost the religious archaeologist, the younger son of the Abrahamic family tree that is taught how to reconnect to their roots and bring forth an organized faith based on the shared one God of the Abrahamic faiths. Remember that the Prophet Muhammad (PBUH) died just about six hundred years after Jesus died and resurrected. God in his mercy works to speak to all his faithful in times throughout history, not to negate what went before, but to continually bring people back to his original Word as God revealed himself to Adam and his descendants. Islam did not spring up as a disagreement with what went before, far from it. Islam was the bringing to life a branch of people in an arid point in their history where their connections back to God had been obscured, so God brought the Prophet (PBUH) forth to revive and glorify these connections to Adam and his successors in salvation history.