We study nonlinear stability of steady isolated vortices in two-dimensional compressible media in a uniformly rotating reference frame. First, we consider a vortex with a linear profile of velocity. Its behavior can be completely described by a quadratically nonlinear system of ODEs. We find that the stability property depends only on one parameter, the ratio of relative vorticity of the vortex to the Coriolis constant. We find the domain of this parameter ensuring nonlinear stability. Further, we consider a more general class of isolated steady vortices, containing decaying at infinity and compactly supported vortices as particular cases. At every point of the plane, this isolated steady vortex can be approximated by a solution with a linear profile of velocity. Thus, at every point of the plane, there arises a nonlinear system of ODEs with initial data generated by derivatives of the steady vortex state. It is hypothesized that if at every point the solution to this ODEs system falls in the domain of attraction of an equilibrium, then the steady vortex is nonlinearly stable. We compare this nonlinear stability hypothesis with Raleigh criterium of linearized stability with respect to radial perturbation and the results of numerical computations. In particular, we find that the rotation has a stabilizing effect.