The index of refraction is an electronic property, directly dependent upon the electric permittivity of the media, and ultimately depends upon the free electron density. As the energy of the photon increases, the interactions with the electrons decrease, so that x-rays are barely retarded at all, though they are scattered by the electrons of the atoms. Of course, the Cerenkov radiation is in the optical range, and so is slowed by the ice by the refractive index.
The net effect is that light travels at c/n in the ice; for ice n=1.3, so for light in ice it travels about 3/4 c.
The muons are created inside the ice when a high energy neutrino is absorbed. The muons which are traveling faster than 3/4 c are the ones which create the Cerenkov radiation; the slower ones will just create ion trails, and not the optical shock waves.
All of the muons interact with charged particles, leading to your question of why they travel faster than light in ice, leading to Cerenkov radiation. Because the ones which generate Cerenkov radiation are traveling at relativistic speeds, this reduces the effective cross section for scattering. They will barely see the electron clouds, but instead scatter of the nucleus.
For those curious as to why light slows down in the ice, I've repeated my previous answer to this question:
Transparent materials (glass, air) transmit images; if the image is distorted or indistinct, we know that the material is altering the coherence of the optical information. That is, what started out at the beginning has not arrived all at the same time. With enough distortion the image is completely lost.
So what is required for a transparent medium to successfully transmit an image? Since light is a physical wave, the transparent medium must preserve the coherence of the phase information of the light. In a typical glass the phase front is slightly slowed while traveling through the glass; this slowing is encoded in the index of refraction, $n = c/v$.
If the material absorbs some frequencies, the material will appear to be colored; a photon that is absorbed (depending on the energy level structure) can be re-emitted, but this will be at (a) a random time later, and (b) in a random direction. No image for this color! There is an exception: stimulated emission, which is the key to building a laser. But this is not how images are transmitted in a passive material.
The process that transmits images can be summed up as Coherent Forward Scattering: Coherent, because otherwise the image integrity is reduced; Forward, because the image is transmitted in this direction, through the material; and Scattering, the remaining available generalized mechanism at the quantum level.
The result is quite like the Huyghen's wavelet model for light transmission: the photons are the waves that are scattered coherently, and because it is coherent, they are able to interfere both constructively and destructively to maintain the coherence of the overall phase front.
It is the interference that slows the phase velocity through through the material; the individual photons continue to "move" at the speed of light, $c$, but the effective motion of the phase front is slowed.
Richard Feynman devotes some time to this in his lectures on QED: The Strange Theory of Light and Matter