The interaction potential of molecular hydrogen physisorbed on a graphene sheet is evaluated using the ab initio-based periodic dlDF+Das scheme and its accuracy is assessed by comparing the nuclear bound-state energies supported by the H2(D2/HD)/graphite potentials with the experimental energies. The periodic dlDF+Das treatment uses DFT-based symmetry-adapted perturbation theory on surface cluster models to extract the dispersion contribution to the interaction whereas periodic dispersionless density functional (dlDF) calculations are performed to determine the dispersion-free counterpart. It is shown that the H2/graphene interaction is effectively two-dimensional (2D), with the distance from the molecule center-of-mass to the surface plane and the angle between the diatomic axis and the surface normal as the relevant degrees of freedom. The global potential minimum is found at the orthogonal orientation of the molecule with respect to the surface plane, with an equilibrium distance of 3.17 Å and a binding energy of −51.9 meV. The comparison of the binding energies shows an important improvement of our approach over the vdW-corrected DFT schemes when we are dealing with the very weak H2/surface interaction. Next, the 2D nuclear bound-state energies are calculated numerically. As a cross-validation of the interaction potential, the bound states are also determined for molecular hydrogen on the graphite surface (represented as an assembly of graphene sheets). With the largest absolute deviation being 1.7 meV, the theoretical and experimental energy levels compare very favorably.