Christianity didn't exactly moderate or accept criticism on its own. It just lost. From the fall of the western Roman empire until the reformation it set the rules and enforced them, through violence if need be. The reformation then led to 1 1/2 centuries of total war during which the church tried to maintain its doctrinal monopoly. It lost.
Westphalia set up the supremacy of the state over religion. Christianity simply wasn't given a choice on the matter. Everything that followed after Westphalia (secularism, liberalism) was simply its consequences thought further and the church (es, at this point) being unable to stop it.
The muslim world is experiencing this conflict, too, roughly since 1920, with iberalisation attempts in Iran and Afghanistan (twice, for the latter), Arab nationalism/socialism in Egypt, Libya, Syria, Iraq, Yemen, Algeria had its own thing going...
Difference is, this same conflict playing out in the Arab world had the opposite result. The states lose, the madrassas win. The only reason there isn't a fundamentalist muslim superstate from Iraq to Morocco is that American bombs keep blowing up the people who are trying to build it.
There are some very old differences between mainstream Christian and mainstream Muslim philosophy, as the long conflict between pure faith and faith through natural philosophy was won by the latter in Christianity (over the course of 1200 years from Augustine of Hippo to Kepler, mind you), while that same conflict in Islam was won by the former between ca. 900 AD and 1300 AD. There is also something to be said for Christianity being quantifiably more liberal than Islam essentially since its inception. Not liberal by any post Westphalian definition of that term, but relatively more liberal than Islam.
But modern Christianity didn't become what it is through choice.
Rather, it was defanged because states forced it to submit with cannons and gunfire.