Where does Rivalry between Christianity and Islam originate?

The same reason that christianity invented antisemitism, as well as the internal ‘rivalry’ among christian sects, in general catholicism vs protestantism: a mere difference in beliefs.

If you’re only looking for biblical references, Muslims would trace the divergence to Abraham’s sons Isaac and Ishmael. In the Judeo/Christian telling of the story, it was Isaac’s lineage that received God’s favor, eventually leading to Jesus (Ishmael and his lineage basically drop off the map in the biblical account). Muslims trace their lineage through Ishmael. They don’t discount Isaac’s lineage, but believe that God used both to give rise to prophets.

Well, the claim of both to be the ultimate revealed religion doesn’t help.

@thorninmud has it as far as origin myth. Isaac/Ismael.

The Christians believe that Jesus marked the end of the “prophets”. Christians can still “prophesy”, if God wants to speak through them, but since the Word is now in the Bible, there’s no need for another Elijah.

The Muslims believe that Mohammad was the last prophet. Jesus is a prophet in Islam, but not the son of God, any more than all of us are sons and daughters of God.

Historically, the Crusades and the Arab-Byzantine Wars did a lot of damage. Biblically, what thorn said.

@ebasboy you wanted the references from the Bible where Muslims and Christians came into being. I see the rivalry start when Hagar becomes pregnant (she despises her mistress) and again during the weaning party (when Ishmael makes fun of Isaac). In fact I’d say that rivalry might be a mild way to put it, because Ishmael was the first born, and now his little brother is going to be the heir. Then it didn’t help when Ishmael and his mother were permanently banned from the camp and put outside.
The biblical order of events:

Genesis 17:5
No longer will you be called Abram; your name will be Abraham, for I have made you a father of many nations (the original prophesy).

The child Ishmael was born first, the natural way because Sara was Abraham’s only wife, and she was very old: Gen. 1:16 Now Sarai, Abram’s wife, had borne him no children. But she had an Egyptian slave named Hagar; 2 so she said to Abram, “The Lord has kept me from having children. Go, sleep with my slave; perhaps I can build a family through her.”

Hagar Sara’s slave girl becomes pregnant with Ishmael through Abraham. When she discovers she’s pregnant, she despises her mistress, and Sara mistreats her. She runs away, but God tells her to go back and submit to her mistress because she is carrying a child that will become the father of a great nation and says this about him:
11“You are now pregnant
and you will give birth to a son.
You shall name him Ishmael,[a]
for the Lord has heard of your misery.
12 He will be a wild donkey of a man;
his hand will be against everyone
and everyone’s hand against him,
and he will live in hostility
toward[b] all his brothers.” (Gen. 16 1:16)

The second child, Isaac was born through a prophesy- a miraculous event because Sara was too old to bear children anymore (Gen 18:1–15).

Isaac was born (Gen 21:1–7).

On the day when Isaac was weaned, they held a big party. Ishmael made fun of his little brother and he and his mother were sent away permanently. (Gen 8–21).

With the finalization of the Quran and Sunnah containing the harsh anti-Jewish and anti-Christian Medina suras abrogating (overriding) the more gentle earlier Meccan suras.