We use rotational excitation spectroscopy with a scanning tunneling microscope to investigate the rotational properties of molecular hydrogen and its isotopes physisorbed on the surfaces of graphene and hexagonal boron nitride (h-BN), grown on Ni(111), Ru(0001), and Rh(111). The rotational excitation energies are in good agreement with Delta J = 2 transitions of freely spinning p-H-2 and o-D-2 molecules. The variations of the spectral line shapes for H-2 among the different surfaces can be traced back to a molecular resonance-mediated tunneling mechanism. Our data for H-2/h-BN/Rh(111) suggest a local intrinsic gating on this surface due to lateral static dipoles. Spectra on a mixed monolayer of H-2, HD, and D-2 display all three J = 0 -> 2 rotational transitions, irrespective of tip position, thus pointing to a multimolecule excitation, or molecular mobility in the physisorbed close-packed layer.