Ocean eddies and fronts affect surface stress via two mechanisms: (1) ocean surface currents altering the relative motion between air and sea and, hence, the stress fields and (2) ocean sea surface temperature (SST) gradients forcing changes in stability and near-surface winds. In this paper, we quantify the first effect and how it impacts Tropical Instability Waves (TIW) in the eastern Pacific. High-resolution satellite data and a regional coupled model are used to distinguish between stress changes due to the surface currents and those due to the changes in stability and near-surface winds. It is found that both mechanisms affect the surface stress curl, but they do so at different latitudes, allowing for their effect on Ekman pumping to be distinguished. The Ekman pumping due to the surface current effect alone, leads to significant damping of the TIWs. In terms of the eddy kinetic energy, the inclusion of surface current in the stress leads to decay with an e-folding time comparable with the period of the TIWs. It is, thus, an important damping mechanism to be included in ocean and coupled ocean-atmosphere models.