A thought experiment.
I have a box of fixed dimensions, inside this box I have molecules of $H_2$ at constant temperature. I'll be gradually increasing pressure inside the box and the temperature shall remain constant throughout the process (we don't consider change in temperature at all).
How will this affect the Root-mean-square speed of gas molecules inside the box?
It won't. The rms speed of an ideal diatomic gas is given by $(3kT/m)^{1/2}$, where $T$ is the temperature and $m$ the molecular mass.
Were you thinking about non-ideal effects? These will start to become important if you increase the pressure enough. The details would depend on the interaction potential between the molecules. At (relatively) low densities these tend to be attractive, but then become repulsive at very high densities. The net effect is often written in terms of a "virial expansion". $$\frac{P}{kT} = n ( 1 + Bn + Cn^2 + ...),$$ where $n$ is the gas number density. For hydrogen I believe the $B$ coefficient is positive, therefore for a fixed temperature and number density, the pressure is higher. However, if you fix it so that the temperature of the gas is kept constant then so is the rms speed unless you make the gas so dense that one has to consider departures from the Maxwell-Boltzmann distribution in the form of Bose-Einstein or Fermi-Dirac statistics.
