How did Islam separate from Christianity?

Answer 1

Islam did not break up from Christianity, it came much later.

Answer:

The teachings of Jesus were the same as all the other Prophets of God: Obedience to the will of the One God. So the REAL Jesus was teaching an earlier form of 'Islam'. That's what all the Prophets of God taught taught.

So "Christianity" - the religion that Paul (Saul of Tarsus) invented - broke away from Islam. Not the other way around.

Answer 2

The question as put is unanswerable.

In order for x and y to separate, x and y must at one point have been indistinguishable or mixed. Islam did not "come from" Christianity nor was it founded by Christians. This is very much unlike Christianity's separation from Judaism where the early Christians were Jews and considered themselves Jews until they and the remaining Jewish community could no longer agree on Jesus' nature.

Most of the first Muslims, including Mohammed himself, were born as henotheists believing Arabian religions that revered Allah as the most powerful god among a pantheon of gods including Hubal and 'Uzza. This was nothing close to Christianity. The development of Islam under Mohammed and the Sahaba (Apostles) actually brought Arabians closer to Christianity, although it never brought them close enough to be confused for being Christians by themselves, by Christians, nor by non-Muslims and non-Christians.