Magnetic reconnection in curved spacetime is studied by adopting a general-relativistic magnetohydrodynamic model that retains collisionless effects for both electron-ion and pair plasmas. A simple generalization of the standard Sweet-Parker model allows us to obtain the first-order effects of the gravitational field of a rotating black hole. It is shown that the black hole rotation acts to increase the length of azimuthal reconnection layers, thus leading to a decrease of the reconnection rate. However, when coupled to collisionless thermal-inertial effects, the net reconnection rate is enhanced with respect to what would happen in a purely collisional plasma due to a broadening of the reconnection layer. These findings identify an underlying interaction between gravity and collisionless magnetic reconnection in the vicinity of compact objects.