Yes, gravitational waves travel at the speed of light. At least according to Einstein’s theory of General Relativity. It would be very difficult to test directly. When we had Maxwell’s Equation predicting the speed of light based on measurable parameters of free space and means of direct measurement of the speed of light (stellar aberration, moons of Jupiter, spinning mirrors, and laser interferometry all give similar results) act to reinforce confidence in Maxwell’s predictions.
It would be nice to have a way of measuring the speed of gravity. I believe that the precession of the perihelion of Mercury is confirmation of the speed of gravity, but my command of General Relativity is nowhere near good enough to really understand it.
I thought Einstein suggested that gravity wasn't a wave at all, but a distortion in space-time. But my understanding of General Relativity is probably worse than yours.