  The "gravitational baryogenesis" scenario is extended to generate both baryon
and dark matter asymmetries, in the matter dominated era corresponding to
post-inflationary reheating. A minimal extension requires a singlet fermion X
for dark matter and a singlet scalar S. With two or more hidden sector
fermions, the scenario can lead to nucleon decay into dark matter with a
lifetime of order 10^{34-36} yr, which is relevant for current or future
experiments. The correct multi-component relic density can be obtained if dark
matter fermions couple to a sub-GeV vector boson that weakly interacts with the
Standard Model through mixing. The typical inflationary scale in the scenario
is of order 10^{16} GeV which suggests that tensor mode perturbations could
potentially be within observational reach.
