You can't just use the ratio of Boltzmann factors to calculate the
probability of an atom being ionized, since there's a continuous
infinity of ionized states that you would have to sum over.
Instead you have to use the Saha equation, which can be derived
from the law of mass action, considering the ionization to be
a chemical reaction. A Boltzmann-like factor does enter the
equation, but the density of the system also enters. The bottom
line is that it's usually much easier to ionize an atom than to
put it into an excited state. Fundamentally, it's an entropy thing.