We examine how the post-processed content of molecular hydrogen (H$_2$) in galaxies from the TNG100 cosmological, hydrodynamic simulation changes with environment at $z!=!0$, assessing central/satellite status and host halo mass. We make close comparisons with the carbon monoxide (CO) emission survey xCOLD GASS where possible, having mock-observed TNG100 galaxies to match the surveys specifications. For a representative sample of host haloes across $10^{11}!lesssim!M_{rm 200c}/{rm M}_{odot}!<!10^{14.6}$, TNG100 predicts that satellites with $m_*!geq!10^9,{rm M}_{odot}$ should have a median deficit in their H$_2$ fractions of $sim$0.6 dex relative to centrals of the same stellar mass. Once observational and group-finding uncertainties are accounted for, the signature of this deficit decreases to $sim$0.2 dex. Remarkably, we calculate a deficit in xCOLD GASS satellites H$_2$ content relative to centrals of 0.2--0.3 dex, in line with our prediction. We further show that TNG100 and SDSS data exhibit continuous declines in the average star formation rates of galaxies at fixed stellar mass in denser environments, in quantitative agreement with each other. By tracking satellites from their moment of infall in TNG100, we directly show that atomic hydrogen (HI) is depleted at fractionally higher rates than H$_2$ on average. Supporting this picture, we find that the H$_2$/HI mass ratios of satellites are elevated relative to centrals in xCOLD GASS. We provide additional predictions for the effect of environment on H$_2$ -- both absolute and relative to HI -- that can be tested with spectral stacking in future CO surveys.
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