They won't become obsolete. Infantry did not become obsolete because of tanks. Tanks did not become obsolete because of Helicopters. Helicopters did not become obsolete because of fighter Jets. Fighter Jets will not become obsolete because of spaceships.

Each of these types of forces serves a different, vital role in war. You cannot capture and hold territory without infantry. You cannot bombard an area without artillery and air. Most relevant to your question, you cannot control the sea lanes and shipping without ships.

Even if we get a bunch of spaceships with orbital bombardment capabilities, they cannot defeat piracy. They alone cannot control territory. You still need a navy if you want to control the oceans. You can certainly bombard a navy from space and destroy it, but then you just have an ocean with no one controlling it.

That being said, control over space resources like asteroids and small moons will bring in trillions of dollars in revenue in today's dollars. It will certainly be the next big stage for conflict. It won't erase the importance of a blue water navy though. It'll simply be in addition to a blue water navy.