It doesn't necessarily mean that spacetime cannot be fundamental, as there's no rule that says that space itself must emerge from some sort of underlying medium (eg the way fabric is made of threads).
Spacetime is warped by mass but no mass is transmitted. So what exactly is being carried along?
The distortion of the fabric of space-time ripples throughout the entire universe at the speed of light, and spreads it's effect proportional to the inverse square related to distance.
In any case, mass is not the only thing that warps spacetime. The components of the stress–energy tensor [1] include energy (which includes mass) density, energy flux, momentum density, and momentum flux, the latter of which is composed of pressure and shear stress components.
[1] https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor
If so, for now we believe that gravity is simply the result of the warping of spacetime, with no known carrying particle. That's kind of the hurdle between uniting gravity with the quantum world.
If say, it turns out that there is such a thing as a graviton - which carries gravity, then gravitational waves would be a bunch of gravitons.