While Christianity has modernized Islam has not
Exactly, modernization for Christianity meant opening up public debate. Modernization in Islam at present, however, is about closing public debate in favor of koranic literalism.
The Muslim world was westerning up until the late 1970s when they collectively gave up and decided to retreat into a kind of weird idealized traditionalism (full of non-traditional things like hijab, very much a late 20th century invention).
Essentially, koranic literalism is the entire foundation of modern islam and it can't really withstand much intellectual scrutiny and people are left with choices,
accept that your religious tradition is just one of many or try to eliminate other traditions
accept that holy texts and science don't need to agree or try to ignore science
accept that holy texts are bound to the time and context of their creation or try to preserve the time and context of their creation
Essentially the muslim world collectively is going to the second option in all these cases