Using molecular dynamics simulations of an atomistic water model, we study the interfacial hydrodynamic slippage of water at various hydrophobic surfaces, both organic (silane monolayers) and inorganic (diamondlike and Lennard-Jones models). The measured slip lengths range from nanometers to tens of nanometers. Slip lengths on different surfaces are found to collapse nearly onto a single curve as a function of the static contact angle characterizing the surface wettability, thereby suggesting a quasiuniversal relationship. This dependence is rationalized on the basis of a simple scaling description of the fluid-solid friction at the microscopic level. The link between slippage and water depletion at hydrophobic surfaces is clarified. These results shed light on the controversy over experimental measurements of the slip length at smooth hydrophobic surfaces.