There are a few varsity sports you can walk-on for.
The biggest walk-on sport at Navy would be their sailing team (both offshore and dinghy). From what I hear, it is a very laid back atmosphere, but will consume a large portion of your time (3:45pm to 7:00pm/7:30pm each day, and most weekends from the start of the year to thanksgiving). Same hours in the spring (but less weekends) from after spring break to graduation. You will also participate in an ocean race, in place of one of your summer training cruises. For the offshore team, most walk-ons will start as junior varsity for the fall semester, and then get integrated into the varsity team in the spring in prep for the summer cruise. For the dinghy team, most walk-ons start as crew. For both sides of the team, its an excellent way to build your leadership skills, and your connections as well!
The other sport that comes to mind is crew. 5:00am workouts on the water, and afternoon sessions are the norm. A sport that definitely keeps you in shape!
If you struggle academically though, you may want to reconsider as it will be a large time commitment on your end (most varsity sports will be). However, it you can manage your time well, its well worth it! Being on a varsity team definitely has its perks!
Please keep in mind, just because these sports take walk-ons, doesn't mean you won't have to earn your spot on the team. You'll probably still have to go through try outs and survive a round or two of cuts, before actually walking on.