Renormalization group analysis is applied to the two-dimensional Navier-Stokes vorticity equation driven by a Gaussian random stirring. The energy-range spectrum C-Kepsilon (2/3)k(5/3) obtained in the one-loop approximation coincides with earlier double epsilon expansion results, with C-K = 3.634. This result is in good agreement with the value C-K = 3.35 obtained by direct numerical simulation of the two-dimensional turbulent energy cascade using the pseudospectral method.