  Robust knowledge of molecular gas mass is critical for understanding star
formation in galaxies. The H$_{2}$ molecule does not emit efficiently in the
cold interstellar medium, hence the molecular gas content of galaxies is
typically inferred using indirect tracers. At low metallicity and in other
extreme environments, these tracers can be subject to substantial biases. We
present a new method of estimating total molecular gas mass in galaxies
directly from pure mid-infrared rotational H$_{2}$ emission. By assuming a
power-law distribution of H$_{2}$ rotational temperatures, we can accurately
model H$_{2}$ excitation and reliably obtain warm ($T\!\gtrsim\!100$ K) H$_{2}$
gas masses by varying only the power law's slope. With sensitivities typical of
Spitzer/IRS, we are able to directly probe the H$_{2}$ content via rotational
emission down to ~80 K, accounting for ~15% of the total molecular gas mass in
a galaxy. By extrapolating the fitted power law temperature distributions to a
calibrated \emph{single} lower cutoff temperature, the model also recovers the
total molecular content within a factor of ~2.2 in a diverse sample of
galaxies, and a subset of broken power law models performs similarly well. In
ULIRGs, the fraction of warm H$_{2}$ gas rises with dust temperature, with some
dependency on $\alpha_\mathrm{CO}$. In a sample of five low metallicity
galaxies ranging down to 12+log[O/H]=7.8, the model yields molecular masses up
to ~100 times larger than implied by CO, in good agreement with other methods
based on dust mass and star formation depletion timescale. This technique
offers real promise for assessing molecular content in the early universe where
CO and dust-based methods may fail.
