Latex paint is water based and it is thinned all the time with water to spray it. The trick is to do it in layers since so much water is needed to get it thin enough to spray. For a door, I wouldn't go through the effort.
As mentioned Floetrol will help reduce brush strokes and give you a smoother finish, but it's mostly a waste of money and not needed for nice results.
When I paint a door with latex, I use a four inch roller and a Best quality smooth finish roller. I do it in two coats since it's so easy to create drips on a door. Paint it fairly dry the first time and don't worry to much if its thinner in some areas then others. Then after an hour, do it again, also as dry as you can keep the roller and still get paint on the door.
Then use the softest 1 1/2 inch brush that you can find. Soft is the secret. Compare everything on the rack, you'll find one that is a lot better then the others. Again, paint it as dry as you can so you avoid creating runs.
I always paint doors while on the hinges with a length of paper under the door. Sometimes I'll use a tarp, but only if I need the tarp for other things.