By using an improved approximation scheme to deal with the centrifugal (pseudo-centrifugal) term, we solve the Dirac equation for the generalized Morse potential with arbitrary spin-orbit quantum number κ. In the presence of spin and pseudospin symmetry, the analytic bound state energy eigenvalues and the associated upper- and lower-spinor components of two Dirac particles are found by using the basic concepts of the Nikiforov-Uvarov method. We study the special cases when κ = ±1 (\documentclass[12pt]{minimal}\begin{document}$l= \widetilde{l}=0,$\end{document}l=l̃=0, s-wave), the non-relativistic limit and the limit when α becomes zero (Kratzer potential model). The present solutions are compared with those obtained by other methods.