Muslims worship the God of Abraham, as do Christians and Jews. Islam was seen as a continuation of the Abrahamic faith tradition, not a new religion.
There is, it's often said, no separation of church and state in Islam. And, historically speaking, this is more or less true.
There was no clear separation of church and state anywhere in the ancient world. High priest and ruler were closely identified (and sometimes identical) in ancient Israel and Judah.
Pre-Islamic Iran was officially Zoroastrian. Augustus was not only the emperor of Rome but "pontifex maximus" or high priest — a title later inherited by the popes. The Council of Nicea, which hammered out the Nicene Creed, was convened by the Emperor Constantine. Virtually all Chinese emperors until the fall of the Qing Dynasty in 1912 were called the "Son of Heaven," claiming to rule under "the mandate of Heaven." The doctrine of "the divine right of kings" survived in Europe until the early modern period.
In mediaeval Christendom, rulers routinely demanded veto power over the appointment of bishops and even the right to choose bishops directly. Kings and other secular authorities, in fact, continued to exercise such rights as late as the second half of the 19th century.
Many countries — by no means limited to Muslim ones — have official state churches or state religions. The British monarch, currently Elizabeth II, bears the constitutional title of Supreme Governor of the Church of England, and nominations for bishops in that church are submitted to the office of the Prime Minister, who vets them on behalf of the ruler. Several of Britain's colonies in the New World also had official churches. Connecticut only disestablished the Congregational Church in 1818; Nova Scotia still recognized the Church of England as its official faith until 1850.
Christianity, though, had one distinct advantage in conceiving the separation of church and state: Constantine's intervention in Christian theology was a novelty (and a bad one). And it wasn't until 380 A.D. that Theodosius I, following universal ancient practice — if not the teachings of Christ — officially adopted Trinitarian Christianity as the imperial religion.
Jesus, however, had held no political office, and, in fact, his followers had spent several centuries struggling against often fierce state persecution. Augustine, a leading Christian theologian, laid the foundation for the separation of church and state when writing his "City of God," describing the church as transcending any state.
By contrast, after his "hijra" (exodus) from Mecca to Medina in 622 A.D., Muhammad became the head of a small Muslim city-state. No longer only a religious leader, "a voice crying in the wilderness," he decided questions of war, inheritance, contract law and the like. Thus, he furnished his followers with an authoritative model of religion and politics that finds no parallel in the life of Christ, but broadly parallels Moses.
In that light, can Muslims ever learn to separate "church" and "state"? Many not only can, they already have.
As illustrated above, even Christendom took many centuries to achieve religiously neutral politics. The European wars of religion following the Protestant Reformation, which lasted from roughly 1524 until at least 1648, certainly played a role: When the warring factions finally realized that they couldn't forcibly convert or kill everyone, they had little alternative but to learn to live with one another, however grudgingly.
Judaism, fostered and sustained for centuries by a theocracy, learned to flourish without such support when Jews were scattered after the destruction of Jerusalem and its temple in 70 A.D. — although it's debatable whether today's Israel is a fully secular state even now.
Islam long enjoyed the patronage of powerful states such as the Abbasid Dynasty, the Fatimids (based in Egypt), the Ottoman Empire (based in Turkey), the Safavids (Iran), and the Mughals (India). Islamic governments collected religious taxes, built mosques and systematically sought to promote the faith.
Islam is still the official religion in many countries, from Morocco to Malaysia, from Egypt through Iran to Pakistan and Bangladesh. Competition from other faiths is often severely circumscribed, if not altogether prohibited.
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But, like Jews, Muslims increasingly live as minorities in societies where they can never realistically hope for formal support of the state. It is in such places, very possibly, that the future of Islam is being worked out — a future that will, necessarily, have to come to terms with religious pluralism and evolve its own institutions, privately funded and privately governed. Some Muslims dislike this. Many, though, welcome it, and are adapting very well to these new realities.
The challenge of their new environment may actually be healthy for Muslims and for their faith. The competitive religious marketplace in the United States has played an important role in creating a vigorous, independent American religiosity that the official Church of England and the tax-supported churches of Austria and Germany, for example, have long since ceased even to dream of.