Renormalization constants (Z-factors ) of vector and axial-vector currents are determined nonperturbatively in quenched QCD for a renormalization group improved gauge action and a tadpole-improved clover quark action using the Schrödinger functional method. Nonperturbative values of Z-factors turn out to be smaller than 1-loop perturbative values by O(15%) at a lattice spacing of a-1[approximate] 1 GeV. The pseudoscalar and vector meson decay constants calculated with the nonperturbative Z-factors show a much better scaling behavior compared to previous results obtained with tadpole-improved one-loop Z-factors. In particular, the nonperturbative Z-factors normalized at infinite physical volume show that the scaling violations of the decay constants are within about 100p to the lattice spacing a-1~1 GeV. The continuum estimates obtained from data in the range a-1~ 1–2 GeV agree with those determined from finer lattices (a-1~2-4 GeV) with the standard action.