The method of moving frames (MMF), introduced for conservational laws (J. Sci. Compt., 53(2), 268-294, 2012) and (an)isotropic diffusion equations (J. Sci. Compt., 59(3), 626-666, 2014), is considered on rotating arbitrary curved surfaces to propose a novel two-dimensional high order scheme for the shallow water equations (SWE) on the rotating earth. In addition to the existing components of the MMF, the moving frames are adapted to represent the velocity vector on rotating surfaces, or generally relative velocity, for the SWE on the rotating earth. Error analyses show that all the error generated by the moving frames can be made negligible compared to the differentiation error, leading to an attractive high-order scheme without adding more schematic complexity. Formulated in the context of discontinuous Galerkin method, the proposed model is validated against the standard tests for the spherical SWE taken from the Williamson test suite. The DG-MMF SWE model is shown to exhibit an exponential convergence and to demonstrate the consistency and stability on ellipsoidal geometries of various curvature.