What is the refractive index of a gas-like system solely consisting of protons? I know that the permittivity of a medium is somewhat dependent on the density of electrons orbiting protons. What if we have a gaseous system with no electrons consisted of protons like H+ or He++? What would be the refractive index then? Is it equal to unity just like the permittivity of a vacuum because of the lack of electrons?    
 A: It would depend on the density of the protons, $n_p$. What you describe is a charged plasma and the penetration of electromagnetic waves into a plasma (the Debye length, I believe)  depends strongly on density. That there are no electrons present is not a problem, since protons can be accelerated by, and therefore absorb electromagnetic waves. However, their mobility is lower, owing to their greater mass.
To calculate the index of refraction, you would first get the dielectric constant, which is done in these notes (specifically, here). The calculation is lengthy, but basically you assume an ansatz for the polarization of the plasma to be $\mathbf{P} = \mathbf{P}_0 e^{i \omega t}$ which you then substitute into an oscillating electric field $\mathbf{E} = \mathbf{E}_0 e^{i\omega t}$. The polarization oscillates at the same frequency because the protons aren't bound (there is no "spring constant"), and so it's purely forced oscillation. If you put this into the Maxwell's equations you find that the dieletric constant varies with $\omega$ and goes to zero at resonance, which is when the incident field oscillates at the plasma frequency:
$$
\omega_p = \sqrt{ \frac{n_p e^2}{\varepsilon_0 m_p^2} }
$$
