Not only will the discovery of gravitational waves confirm an important prediction of general relativity, but it will be the first test of GR in the strong gravity regime. So far, all the tests of GR have been in the weak gravity regime, where Newton's law of gravity is approximately correct. The precession of Mercury's perihelion, gravitational redshift, the gravitational deflection of light, and the measurement of frame dragging by Gravity Probe B have all been probing similar regions of parameter space --- slow velocities (compared to the speed of light) and weak gravity (i.e. far from the event horizon of a black hole).
If the rumors are true that LIGO has detected the merger of two black holes, the observed waveform will be the first test of the predictions of general relativity in the strong gravity limit. This is the most interesting limit because this is where physics diverges the most from Newtonian gravity. Moreover, it's much easier to detect deviations from general relativity in this limit.
LIGO's detection won't be the only test of GR in the strong gravity limit for too long, however. The Event Horizon Telescope has been trying to directly observe the event horizon of the supermassive black hole at the center of the Milky Way (or its shadow, anyway) and should have its first images within a few years.