We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude $\theta$ and of the number of vortices N. By using the integrals of motion, we view the system in a suitable corotating frame where the polygonal vortex configuration is at rest. Then after a sufficient criterion due to Dirichlet, the stability ranges are the $\theta$-intervals for which the Hessian of the Hamiltonian---evaluated at the equilibrium configuration---is positive or negative definite. We find that the stability intervals coincide with those for linear stability determined by Polvani and Dritschel [J. Fluid Mech., 255 (1993), pp. 35--64]. For $N=3$ we recover the result previously established by Pekarsky and Marsden [J. Math. Phys., 39 (1998), pp. 5894--5907].