We perform a numerical test of a relativistic heavy quark(RHQ) action, recently proposed by Tsukuba group, in quenched lattice QCD at a similar or equal to 0.1 fm. With the use of the improvement parameters previously determined at one-loop level for the RHQ action, we investigate a restoration of rotational symmetry for heavy-heavy and heavy-light meson systems around the charm quark mass. We focused on two quantities, the meson dispersion relation and the pseudo-scalar meson decay constants. It is shown that the RHQ action significantly reduces the discretization errors due to the charm quark mass. We also calculate the S-state hyperfine splittings for the charmonium and charmed-strange mesons and the D-s meson decay constant. The remaining discretization errors in the physical quantities are discussed.