My experience with the military is exclusively land based but I've got to believe that an exercise where even the remote possibility of the sequence of events necessary for the sort of "mistake' that you say happened, off the coast of NYC's immediate suburbs, directly in an international flight path where the traffic is so heavy that you can stand on the beach and see as many as a dozen commercial craft in the air, is beyond the scope of what the Navy plans.
Let's not even consider the number of people who would know about it...in the hundreds at least. I guess though that we can expect every 19 year old Seaman on those ships to shut up just because.
One clue here is that Flight 800 was not supposed to be flying that low -- it was supposed to be flying at 16,000 feet or more. It was ordered to descend to make way for a northbound USAir flight into Providence, Rhode Island (flying in a path that was perpendicular to Flight 800's path) that was running behind schedule that night and should not have been in the area.
I know I'm just speculating here -- but the facts as I understand them indicate the very real possibility that Flight 800 was brought down simply because a U.S. Navy target drone got too close to it.