I'm not aware of one. You might try looking it up on the internet. A number of hotels have pools (albeit small for swimming) -- mostly outdoor although maybe some have an indoor pool. Check TripAdvisor or some other site for info on hotels.
I was looking at one of the house rental sites and it mentioned something about the Arundel Swim Center.
I did a search for it and it appears to be only about 4.5 miles from USNA, just west of many of the hotels that are a little out of the downtown area, in the Riva Rd area.
From their site:
"Welcome to Anne Arundel County’s outstanding public swimming facility. The swim center features a temperature controlled, indoor 50 meter by 25 yard swimming pool with adjacent wading pool, a poolside 17-person spa, two 1-meter diving boards and fully equipped locker rooms with coin operated lockers. All facilities are handicapped accessible. The Arundel Olympic Swim Center is open seven days a week. Lanes are continuously available for lap and recreational swimming. Youth, 12 and under, must be accompanied by a paying adult."
I hope this helps, although I think it is just an indoor facility.
I did a search for outdoor pools in Annapolis and it seems that the Annapolis Parks and Rec. have a municipal outdoor pool:
Truxtun Park Pool, Annapolis, Maryland 21403 and it is just over 3 miles from USNA.