On the one hand, it seems to me that there is a sense in which the confession of Christianity has to include a denial of the truth of Islam. The angel Gabriel announced the birth of Jesus Christ to Mary. Some centuries later he is said to have revealed the Qur'an to Muhammed. If I admit that this latter claim is true, even if I also accept the former, I ought to become a Muslim right now. If I go another day without converting to Islam, it seems to me that I'm denying that this revelation occurred. Either Gabriel came to Muhammed or he didn't. Perhaps I am simpleton in this regard, but I don't see any coherent way out of this dilemma.
So it would seem to me, in my own reflection, that there is no theological relationship between Christianity and Islam (in the sense that there is a theological relationship of Christianity to Judaism, not for example) However, I'm not ready to stand on this claim. Why? Because I look to the Church's teaching, and I don't know what to make of it. Nostra aetate, Vatican II's famous decree on non-Christian religions, has some nice things to say about Islam, but does not get at the question of a theological relationship, of a sense in which the existence of Islam might have meaning for Christianity (and hence for God.)
Lumen gentium 16 presents something harder:
Sed propositum salutis et eos amplectitur, qui Creatorem agnoscunt, inter quos imprimis Musulmanos, qui fidem Abrahae se tenere profitentes, nobiscum Deum adorant unicum, misericordem, homines die novissimo iudicaturum.
But the plan of salvation also embraces those who know the Creator, among whom first are the Muslims, who profess to hold the faith of Abraham, with us adore the one, merciful God, and will judge the human race on the last day.
The "propositum salutis," the plan, or design of salvation, which we read to be God's plan, seems to include Islam, at least according to Lumen gentium. As I described above, I'm not sure how to understand this myself, but it is what the Church seems to say.
Amplector is a rather interesting and suggestive verb in this text: amb, "around" plus plecto from the Greek, πλεκω, to twine, braid, or weave.