Not Islamic. A fundamental reason Christianity could reform at all stems from the written teachings of Christ. (Ironic sure and this will sound preachy for a moment, but hopefully to be resolved at the end.)
When accused of plucking grains of harvest on the Sabbath which was "working on the Sabbath", Jesus queries [who would leave their donkey in a pit if it fell in one on the Sabbath?] ie: The Law was made for man, not man for the law.
When accused of not punishing a woman (accused) of adultery, Jesus queries [He who is without sin may cast the first stone].
Jesus states [I will tear down the temple, and rebuild it in 3 days]. ie: I will change everything you thought you knew about the ways of this world and your beliefs.
There are numerous other statements from Jesus that turns the ancient (and current) law on it's head.
Jesus actually gave the Jews (and any other persons willing to believe) the absolute right to shrug off ALL the notions of human rigid indoctrination and become self-fulfilling in the spirit of the creator.
There are many who did, do, and will twist this "power" to their own means. But in reality Jesus was the essence of life lived through the Spirit of the Creator. We can do exactly the same. When lived through belief in Jesus one finds Faith, Charity, and Love as the fruit of practicing living through respect of his salvation of that spirit. But no one is beholden to anything other than themselves and their actions through that spirit. There are no "rules" to believing in the life of Christ. It is seriously as free as free could be, and still be utterly pure of spirit.
All that to say this: I do not think Islam contains such a fundamentally philosophic shattering of ancient ideology as what the Christ represented. His story leaves many confused (including the Jews who carried and birthed his story), and in many ways they remain as static as Islam. In other words, there is no mechanism, no realization for them to point to in the Quran to allow for such a shift of belief. Christians are in no way held to practice the Jewish Law, Islamic law or any other human law and are only directed to respect human laws in their own context; not rigid but perhaps reasonable due to overall human fallibility and logical practice of human disputes.
Without this stunning Crux between the Old testament and the New testament (both very human text, and both very fallible) Christianity would truly be no different than Judaism and/or Islam. It would solely be rigid and difficult to navigate in a dynamic universe. Jesus, in the book written of him, gave us freedom to NOT blindly follow the text surrounding him. Learn it, Know it, Use it... but in the end see the writing for what it is. Life speaking to humans through our own actions. And our actions are inherently fallible and it shows in our understanding of our own writings. Basically you have to be willing to NOT follow the text to fully understand the good that is in it. Islam doesn't much allow for that, and without a guide like Christ, I don't know how that could be reconciled.