Will the Navy have to submerge?
Yes, and come up with stealth platforms difficult to target for surface ships and improved ECM capabilities for everyone.
Will the battle for the control of the seas actually be fought in space?
Absolutely. Both militaries are now capable of shooting down satellites. I expect the pious demands on the part of the UN that space never be "militarized" to be just as effective as the Washington Naval Treaty of 1922, which prevented WWII... :-)