Clathrin-mediated endocytosis is independent of actin dynamics in many circumstances but requires actin polymerization in others. We show that membrane tension determines the actin dependence of clathrin-coat assembly. As found previously, clathrin assembly supports formation of mature coated pits in the absence of actin polymerization on both dorsal and ventral surfaces of non-polarized mammalian cells, and also on basolateral surfaces of polarized cells. Actin engagement is necessary, however, to complete membrane deformation into a coated pit on apical surfaces of polarized cells and, more generally, on the surface of any cell in which the plasma membrane is under tension from osmotic swelling or mechanical stretching. We use these observations to alter actin dependence experimentally and show that resistance of the membrane to propagation of the clathrin lattice determines the distinction between ‘actin dependent and ‘actin independent’. We also find that light-chain-bound Hip1R mediates actin engagement. These data thus provide a unifying explanation for the role of actin dynamics in coated-pit budding.