We consider the 2-body problem in the sphere S-2. This problem is modeled by a Hamiltonian system with 4 degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of 2 degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability results on Hamiltonian systems for demonstrating a series of results that will correspond to the open problems proposed in [4] related to the nonlinear stability of the relative equilibria. Moreover, we study the existence of Hamiltonian pitchfork and center-saddle bifurcations.