Gravitational waves are a prediction of general relativity - an old one. String theories can produce the Einstein Equations via a graviton particle, but I don't know if any of the various versions of this include gravitational waves, or if the waves produced would differ in any experimantally detctable way from the ones predicted by GR.
The current attempts to detect the waves are driven by the GR predictions. connections to string theory would be unforeseen but welcome.
As I understand it, the gravitational wave would be the average of very many gravitons, just as the electromagnetic wave is the average of a great many photons. So we would have to construct a wave detector sensitive enough to resolve quantum levels in the wave amplitudes to prove the existence of gravitons.