Motivated by a recently found interesting property of the dark halo surface density within a radius, \r\$_\max $, giving the maximum circular velocity, \V\$_\max $, we investigate it for dark halos of the Milky Way\textquoterights and Andromeda\textquoterights dwarf satellites based on cosmological simulations. We select and analyze the simulated subhalos associated with Milky-Way-sized dark halos and find that the values of their surface densities, \\\\ensuremathΣ \\\$_\V$$_\max $\, are in good agreement with those for the observed dwarf spheroidal satellites even without employing any fitting procedures. Moreover, all subhalos on the small scales of dwarf satellites are expected to obey the universal relation, irrespective of differences in their orbital evolutions, host halo properties, and observed redshifts. Therefore, we find that the universal scaling relation for dark halos on dwarf galaxy mass scales surely exists and provides us with important clues for understanding fundamental properties of dark halos. We also investigate orbital and dynamical evolutions of subhalos to understand the origin of this universal dark halo relation and find that most subhalos evolve generally along the \r\$_\max $∝ \V\$_\max $ sequence, even though these subhalos have undergone different histories of mass assembly and tidal stripping. This sequence, therefore, should be the key feature for understanding the nature of the universality of \\\\ensuremathΣ \\\$_\V$$_\max $\.