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We describe a novel meshless Galerkin method for numerically solving semilinear parabolic equations on spheres. The new approximation method is based upon a discretization in space using spherical basis functions in a Galerkin approximation. As our spatial approximation spaces are built with spherical basis functions, they can be of arbitrary order and do not require the construction of an underlying mesh. We will establish convergence of the meshless method by adapting, to the sphere, a convergence result due to Thom\'ee and Wahlbin. To do this requires proving new approximation results, including a novel inverse or Nikolskii inequality for spherical basis functions. We also discuss how the integrals in the Galerkin method can accurately and more efficiently be computed using a recently developed quadrature rule. These new quadrature formulas also apply to Galerkin approximations of elliptic partial differential equations on the sphere. Finally, we provide several numerical examples, including the Allen-Cahn for the sphere.