You are using the wrong scientific terminology but you are right in a sense. The speed of light is indeed determined by the environment it travels in. We don't call it medium in relativity or physics, we call it spacetime. Spacetime has a geometry, a 4 dimensional (ignoring quantum gravity) manifold that has 1 time dimension and 3 spatial ones. The simplest spacetime is Minkowski, which is flat. The metric of it, that which describes its geometry (though not its global topology) is:
ds^2 = - c^2 * dt^2 + dx^2 + dy^2 + dz^2
That spacetime means that anyone in an inertial reference frame on it will see any massless particle travel at speed c.
For General Relativity (GR)it is curved spacetime, and the metric is similar to the above but each term can have functions of space and time in front, and mixed terms (also with functions in front) like dxdt dxdy and so on. The functions in front are determined by Einsteins gravitational equations which relate the metric (i.e., the spacetime) to the matter content in the spacetime. The interesting thing about GR for your question is that the speed of massless particles, in a LOCAL reference frame is also c in ALL those cases. It is said that they locally obey special relativity.
So you see, it is the spacetime which determines the speed of light, gravitons, and any other massless particle. And it is always c. So in a sense you are right. But it is even more, spacetime also determines the motion of any test particle (i.e., small enough so you can ignore its own gravitational field), and in fact all test particles move exactly the same way in the gravitational field. They move as free falling particles, in spacetime curves determined by the spacetime metric, and INDEPENDENT of their mass. This is the so called principle of equivalence, the spacetime determines the motions of freely falling particles. It's the same reason in Newtonian gravity astronauts float exactly the same inside their space station orbiting the earth, regardless of how heavy or small they are. GR says the same. (But there's plenty other differences).
Your idea of the medium determining the speed of the waves sort of means in physics that a material medium is involved. It really is not, it is just spacetime. A wireless or electromagnetic medium is a standard term used in wireless and optical communications, an electronic engineering view of electromagnetic propagation. In that case it means the medium, whether air or clouds or glass or fiber, or for cellular wireless the air plus the terrain and objects around from which the waves diffract, reflect, scatter and get absorbed. Those are different concerns and physical media. In physics we are talking mostly, when we talk about c, about empty space, and with gravity possibly other matter around creating that gravity (for cosmological distances plenty of mostly empty space, which however can be curved due to the other matter in the universe. Similarly for the spacetime outside stars, or outside black holes). In all of those the local speed of massless particles is c.
So you are partially right. It is all about the environment in which they travel. Because of the equivalence principle nothing else can make a difference.