Actually, they did not. In fact, it wasn't until mathematicians and other thinkers turned away from the influence of Aristotle that the scientific method was developed. Roger Bacon in the thirteenth century and Johannes Kepler 400+ years later took the first steps out of the ancient thinking into the modern world.
The Christian religion did not give birth to science--that would be overstating the case to the point of silliness, but there are characteristics intrinsic to Judeo-Christianity which turned out to be essential to the development of modern science, characteristics that were lacking in other religions, including ancient paganism. Christianity provided the proper soil.
Once could argue that Aristotle took things off the rails, but before him people had developed empiricism which really is the method at its heart.
I will certainly concede that the modern scientific method came about after rejecting Aristotle’s errors. That took longer than it should have since he was so well regarded.
As you say, and I agree, Christianity was a religion that at least didn’t get in the way of the development (with hiccups, eg Galileo). Other religions, Islam, Bhuddism, Hinduism have mysticism and anti-scientific thinking at their cores.
Nicely put.