I think aside from the original expansion there really isn't much that can be proven one way or another.

Christianity primarily spread in a Roman Empire that already existed (as well as other places) while Islam was spread within a caliphate/empire that was created by it. Muhammad united (politics and military) Arabia and his successors created the Rashidun caliphate that conquered large parts of the Middle East. With both religions generally the initial conversions are relatively peaceful. It took centuries for places like Syria/Egypt to become majority Muslim as Christians were tolerated (and there are still Christians).

Since Islam expanded with Islam as a dominant religion and Christianity had to become the dominant religion its comparing apples and oranges imo.

That being said once Christianity became dominant and tied to the state I would say both religions have a pretty similar history. Tons of nuance here too. Islamic expansion into India was often brutal (although debated) but it wasn't foreign conquerors in say Indonesia and parts of India. Christian expansion into still Pagan Europe was by conquest with the Northern crusades as well as violent crusades against heresy (cathars) but peaceful conversion happened all the time as kings converted for various reasons and the process of becoming christian or Muslim is really depends on where, when and who.