Two-dimensional turbulence generated in a finite box produces large-scale coherent vortices coexisting with small-scale fluctuations. We present a rigorous theory explaining the eta=1/4 scaling in the V is proportional to r(-eta) law of the velocity spatial profile within a vortex, where r is the distance from the vortex center. This scaling, consistent with earlier numerical and laboratory measurements, is universal in its independence of details of the small-scale injection of turbulent fluctuations and details of the shape of the box.