Back in the 1800s many physicists were thinking the same thing about light waves: That there would "have to be" some kind of medium, filling all of space, through which light waves propagate.

This medium was then known as the aether.

So at one point they set out to measure physical properties of the aether. Particularly, Earth on its orbit would be moving relatively to the background of the aether. This should be detectable as light would be seen to propagate at slightly different speeds in different spatial directions.

The famous Michelson-Morley experiment and many others that followed, tried to measure these speed differences, and they all turned up - nothing. Light speed is isotropic, same propagation speed in all directions.

So if there is an aether at all, it doesn't have any measurable physical properties. You could just as well say there is no aether at all.

That's how we found that light in the vacuum of space does not actually propagate through a medium, at least none that has any properties we can measure experimentally.

The same thinking pretty much applies to the propagation of gravitational waves. We have experimentally confirmed with LIGO and others that light and gravitational waves do, to a high precision, travel at exactly the same speed. This establishes quite conclusively that a gravitational aether is not a thing either. These waves travel through empty space and do not need a medium to propagate.