You can blow up all the satellites in space from earth way faster than you could from the moon. I agree with the gravity thing, but for all intents and purposes, the moon might as well be mars as far a war is concerned.

Hitting strategic targets is a time sensitive matter in a war, and there is simply too much distance between the earth and moon for any moon based weapon to be viable. Space based weapons against ground or air targets are largely pointless anyway, as the orbital period at LEO is about 25 minutes, and it takes several minutes for something to go from orbit to ground. So your satellite only passes over your target every 30 minutes, and it take more minutes for your munitions to reach the target once released, and you need a dedicated satellite for each target you want to hit. An ICBM can be anywhere on earth in 30 minutes and any of them can be targeted at any random point you want.

These issues only get exaggerated when trying to fire from the moon. The earth rotates every 24 hours, but it’s going to take days for anything you fire to even reach earth. So you have to fire your munition not at your target, but where your target will be on earth when it’s beneath the moon, when your munition will reach earth. It’s like trying to hit a fly glued to a ball that’s spinning, while you’re on the opposite end of a 2 football fields and you’ve got an air rifle.

Likely any war in space is going to be relegated to controlling area on the body in question. A war between china and the US won’t have a US moon base firing on the china on earth, it’ll have the war on earth, and the moon bases fighting between themselves.