In this paper we have studied the approximate analytical solution of scattering states of the spinless Salpeter equation (SSE) with the Yukawa potential by using an approximation to the centrifugal term that is valid for a short range Yukawa potential. We obtained the energy eigenvalues and the normalized radial wave functions, as well as the scattering phase shifts, which are separately calculated. We comment also on the special case of the Coulomb potential. We have also provided some numerical results for comparing our results with the finite difference method.