I did it for 8 yrs in the Marines. Loved it! I'd still be doing it if I didn't go off to the Army to fly helicopters.
When I joined it was a dual MOS in that you could go either ATC or C-130 navigator. I was only 1 of 7 that qualified for C-130 navigator but I chose ATC. I don't regret it. The first day of class our instructor made it clear that we were in the best MOS the military has to offer. We quickly found out it had one of the highest washout rates also. C-130 navigator actually had the highest around 50%. Anyway over the next 16 weeks we were put through the ringer. Classes on airspace, weather, flight ops, radar and tower simulations. Out of our original 16 we dropped I believe 6. Back then it was at Memphis, now it's in beautiful Pensacola. After graduation I was stationed at Beaufort SC, Okinawa, and Miramar CA. Great experiences working every type of military aircraft imaginable including the Blue Angels and Air Force One. Also worked various GA aircraft as well. I was rated at radar final controller, approach, and got my control tower operator (CTO) in Okinawa. Also a supervisor in tower and radar.
I also did the tactical control out of state of the art portable radar systems. We essentially went into places like South Korea and Thailand during military exercises and took over their ATC system. We taught them radar control as well.
If I were to pick a branch of service I'd go Air Force though. My brother did ATC in the Air Force and would have as well but they would not give me a guaranteed contract so I went Marines. In the Air Force you'd have. Few benefits. First, your basic would be easy instead of the 12 weeks of pain I went through at Parris Island! You'd be pretty much Guarrenteed to go to a base that does a good traffic load. Probably not deploy anywhere. Your all around treatment and living conditions are quite good. Navy you would work some serious traffic as well but you run the risk of going out on the boat for 6 months at a time...unless that's what you want. I went out to the boat for 2 days to see how the Navy does ATC on a ship. Coolest thing in the world. Gotta take a trap/catapult in the COD, FOD walked around Tomcats and Hornets cranking up for the next cycle of ops and finally watched the controllers work their magic in CATCC. Definitely don't go Army. They don't work squat for traffic and it's all helos. Plus you'll find yourself doing things that have nothing to do with your MOS. I flew VIP in Afghanistan and my radio operator in the back was an Army controller. Often they do exercises of setting up and tearing down their portable gear and then rhey never use it. In the Marines we used it. We would go out to an island set up shop with portable approach,tower, TACAN and run C-130 into a 5,000 ft coral runway. Awesome stuff!
Now I don't know if you checked on the asthma thing but they do flight physicals every year. We were controllers but still were considered in flight status. If you go Air Force or Navy you can get back seat rides in jets as well...at least they did when I was in. FAA controllers used to get free commercial jump seat rides for controllers. You just showed them your "pink card" and your in. After 9/11 that all changed that policy. Anyway just make sure you can qualify with asthma. I Guarrentee you won't regret going ATC. Give it 100 % and it will be the most rewarding experience you can do in aviation. Good luck.