The damping of rotations of a hydrogen molecule - modelled by a point quadrupole - due to the excitation of electron-hole pairs in a nearby Cu surface - modelled by a semi-infinite jellium - is calculated. The lifetimes of the rotational states depend on the distance d between the molecule and the surface asymptotically as d⁶. For H₂ physisorbed on Cu the lifetimes are found to be of the order of 10^-9 s, a value 2-3 orders of magnitude too large to explain widths measured by vibrational spectroscopy.