By analyzing the equilibrium shape of vacancy islands on the Ge(001) surface we have determined the surface stress anisotropy, i.e., the difference between the compressive stress component along the substrate dimer rows and the tensile stress component perpendicular to the substrate dimer rows. In order to extract the surface stress anisotropy we have used a model recently put forward by Li et al. [Phys. Rev. Lett. 85, 1922 (2000)]. The surface stress anisotropy of the clean Ge(001) surface is found to be 80+/-30 meV/A(2). This value is comparable to the surface stress anisotropy of the closely related Si(001) surface.