Bullets are much different than large ordnance, like that used in cannon type mechaisms or ship's large guns. Bullets are propelled by a gunpowder that does not explode, but rather burns rapidly, very rapidly. When the projectile strikes an object, the damage is done by the projectile. In large ordnance the projectile carries an explosive charge which detonates, which ultimately causing the damage.
Bullets are quite safe to handle, as long as you don't strike the primer on the end.
Military ammunition is sealed around the primer and around the bullet, the only two possible points of entry for water. Since they are sealed, they could very well remain live for years, at least until the case corrodes through and water contaminates the propellent.
The case is likely made of brass, although some steel casings are known to have been used. I do not know how long it would take for brass to corrode through in saltwater.