Determining the nature of electronic states in doped Mott insulators remains a challenging task. In the case of tetragonal La_{2-x}Sr_{x}NiO_{4}, the occurrence of diagonal charge and spin stripe order in the ground state is now well established. In contrast, the nature of the high-temperature "disordered" state from which the stripe order develops has long been a subject of controversy, with considerable speculation regarding a polaronic liquid. Following the recent detection of dynamic charge stripes, we use neutron scattering measurements on an x=0.25 crystal to demonstrate that the dispersion of the charge-stripe excitations is anisotropic. This observation provides compelling evidence for the presence of electronic nematic order.