All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
It is not permissible for a Muslim to believe in the content of Darwin's theory because it is based on denying what Allah revealed through His Prophet and this is undoubtedly Kufr (disbelief).
Shaykh Ibn 'Uthaymeen said in Fataawa Noor ‘ala Ad-Darb, “This statement is not correct, that the origin of man is an ape, and believing in this is Kufr, as it means denying the Quran. Allah clarified that the origin of creation of man is clay, by the creation of Aadam (Adam), may Allah exalt his mention, who was the father of mankind, from clay; and then Allah made Aadam’s posterity out of the extract of a liquid disdained...”
There should be a distinction here between Kufr and the Kaafir (disbeliever). If a person commits an act of kufr, there may be impediments to declaring him as a Kaafir. If a person’s Islam is proven with certainty, it cannot be overruled except with like certainty.
For more benefit on the conditions for declaring someone as a Kaafir, please refer to fatwas 8106 and 87963.
Allah knows best.