Yes it is. In fact you could technically join all services if you wanted to (of course not at the same time). If you are currently under contract with the Army they will have to release you under the condition that you have been accepted by the Navy. This protects them as if the Navy does not accept you then you have to stay with the Army. Of course there is a form to fill out and has to approved by your commander. If you are not currently under contract with the Army then it should be no problem to join the Navy.
I went from the Marine Corps to the Air Force Reserve and then to the Air National Guard. Believe it or not there are a lot of people who switch services for one reason or another.
