Molecular hydrogen adsorbed on graphene was investigated by analyzing rotational excitation spectra obtained with a gate-tunable scanning tunneling microscope (STM). Through the shift of the rotational excitation energy, the tunability of physisorbed H2 on graphene was evaluated in response to electric fields and surface charging. Since pristine graphene and molecular hydrogen are both inert in nature, direct observation of physisorbed H2 on graphene remains elusive as H2 binds too weakly to flat graphene, even at low temperatures. Here, molecular hydrogen was exposed to wrinkled graphene at 5 K. Localized charges in the wrinkled graphene stabilize the H2-graphene bonding and thus enable observation using gate-tunable STM. Gate biasing modifies the charge carrier density in graphene and the electric field applied to the system, allowing for control of the physisorption of H2 molecules on graphene. The interaction between the molecule and surface is altered significantly as the gate voltage changes from -40 to +40 V, which results in a shift of the J = 0 -> 2 rotational excitation of H2 from 47 to 53 meV. Our theoretical results show that the rotational energy barrier follows a parabolic function of the electric field whose maximum shifts with surface charge. This trend of electric field and charge dependence was experimentally observed by STM. Modulating the electric field and the amount of surface charging offers unique opportunities to control physisorbed molecules, which is reversible and nondestructive.