It is well known since the works of Utiyama and Kibble that the gravitational force can be obtained by gauging the Poincare group, which puts gravity on the same footing as the standard model fields. The resulting theory-Einstein-Cartan gravity-inevitably contains four-fermion and scalar-fermion interactions that originate from torsion associated with spin degrees of freedom. We show that these interactions lead to a novel mechanism for producing singlet fermions in the early Universe. These fermions can play the role of dark matter particles. The mechanism is operative in a large range of dark matter particle masses: from a few keV up to similar to 10(8) GeV. We discuss potential observational consequences of keV-scale dark matter produced this way, in particular for right-handed neutrinos. We conclude that a determination of the primordial dark matter momentum distribution might be able to shed light on the gravity-induced fermionic interactions.