$c$ is not thought of primarily as the speed of light (even in a vacuum) in modern thinking - it's a universal constant that comes to mean the speed of any massless particle. It's existance, as a unique speed which is constant for all observers, is foretold by basic symmetry and homogeneity arguments of special relativity. See my answer to "The reference frame of c ", where I discuss these symmetry arguments.
Experimentally light's speed in a vacuum is observed to transform in this very special way, so this tells us that light is a massless particle, and, because special relativity foretells that the universal speed $c$ that transforms in the special way is unique, we know we're measuring the "right" one when we measure light's speed in a vacuum and declare it to be $c$.
Interestingly, you can turn this argument around and show that, because light's speed transforms in this unique way, we know we're not immersed in any "medium" that interacts with light and which we're not aware of (e.g. "transparent" dark matter) - see here.
Note that if we were immersed in some kind of medium like this (with refractive index 1.00001, say), and Michelson and Moreley did their experiment today, they would get a non-null result. But they still wouldn't get the result foretold by Galilean relativity. So the result would not tell against special relativity, and indeed we'd be able to get good estimates on what the true value of $c$ is, even though our measured light speed would be ever so slightly lower.
Optical mediums "break" the symmetries underlying special relativity and velocities of light in them transform under boosts quite differently from $c$ . The matter of the medium throws up a "preferred" reference frame because the atoms / molecules in it "hold on to the light" for small time intervals in a process of cyclic absorption and re-emission that begets a "slowing of light" and thus an effective refractive index. The delays are nonzero and thus are dilated under boosts, leading to a dependence on frame of the refractive index and the measured speed.