Although we often hear of Christian arguments against Darwinism, we are less informed about Muslim reactions to evolutionary theories: here is an interesting debate where a Muslim scholar defends intelligent design.
Note how creationist theories, constructed for Christian literal fundamentalists, are adapted by Muslim creationists in Turkey, for different audiences.
My answer is yes, on two different levels. First, a believer can
reasonably accept a guided evolution, a process determined and directed
by God. I think this is compatible with Islam or Christianity.
Second, one can believe even in a purely Darwinian (unguided, driven by chance and natural law) evolution and still can be a Muslim or Christian. However this is a very shaky and problematic position. Because the Qur'an -- and the Bible, for that matter -- tells us that the existence of God is evident in what He has created. There are many verses in the Quran telling about God's signs in nature; in animals, plants or the human body. But Darwinism says, "Well, no, there are no such signs, because all of this is created by the blind forces of nature."
In other words accepting Darwinism does not force you to deny God, but forces you to accept that there is no rational evidence from nature to believe in God. It does not make you necessarily an atheist, but it portrays atheism as a perfectly rational, reasonable position. That's why the arch-atheist Richard Dawkins thanks Darwin because "Darwin made it possible to be an intellectually fulfilled atheist."
It is clear that some Islamic scholars adopt Christian arguments on intelligent deisgn versus Darwinism, as an adjunct to their own fundamentalism. Without further knowledge of Islam's reception of Darwinism, it is not clear if this transmission of knowledge is recent or reflects an older relationship between Christian and Islamic fundamentalism.