We study the effects of finite proton mass on the energy levels of hydrogen atoms moving transverse to a superstrong magnetic field B with generalized pseudomomentum K _ { \perp } . Field strengths of order B \sim 10 ^ { 12 } Gauss are typically found on the surfaces of neutron stars , but we also study the regime B \mathrel { \raise 1.29 pt \hbox { $ > $ } \mkern - 14.0 mu \lower 2.58 pt \hbox { $ \sim$ } } B _ { % crit } = 4.23 \times 10 ^ { 13 } Gauss , where the Landau excitation energy of the proton is large . We adopt two different approaches to the two-body problem in strong magnetic field , and obtain an approximate , but complete solution of the atomic energy as a function of B and K _ { \perp } . We show that , for B > > B _ { crit } , there is an orthogonal set of bound states , which do not have any Landau excitation contribution in their energies . The states with very large K _ { \perp } have small binding energy and small transverse velocity , but are nevertheless distinct from the fully ionized states . The final results for the excitation energies are given in the form of analytical fitting formulae . The generalized Saha equation for the ionization-recombination equilibrium of hydrogen gas in the presence of a superstrong magnetic field is then derived . Although the maximum transverse velocity of a bound atom decreases as B increases , the statistical weight due to transverse motion is actually increased by the strong magnetic field . For astrophysically interesting case of relatively low density and temperature , we obtain analytic approximations for the partition functions . The highly excited bound states have a smaller statistical weight than the fully ionized component .