  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
survey's 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.
