Relativity is not based on light. The existence of something that travels at the speed limit, e.g., light just lets us tell some stories about how things work that let us derive some of the equations of relativity with very simple math. Otherwise the path to derive the same equations would be slightly more complicated.
We've discussed this before: .
Gravitational signals also travel at the same speed limit. We don't have a direct measure of that, but as you suspect the whole structure of relativity would fall apart if it weren't true. The slowing of binary pulsars by gravitational radiation confirms this conclusion quantitatively, although indirectly.
Mike W.
(published on 03/13/2015)