You can't call it a navy because those who don't read scifi (almost everyone) won't be able to grok the logic of calling something that doesn't get wet a "navy". Can you imagine trump having to have this explained to him monosyllabically?
Besides...what this is all going to wind up being is just missiles that shoot down satellites and other missiles. And maybe some experimental laser beams pew pew...which is stuff we've already got. It'll be a good long time before there's any need for any sort of warship in space. What's it going to do up there besides be a target for said missiles? Defend moon rocks from Chinese interlopers? Actually, knowing the chinese, they'll probably just boost some dirt from the bottom of the South China Sea up there and declare whatever volume of space around it to be Chinese Territorial Waters.