We present a method to nonperturbatively determine the parameters of the on-shell, -improved relativistic heavy quark action. These three parameters, , , and , are obtained by matching finite-volume, heavy-heavy, and heavy-light meson masses to the exact relativistic spectrum through a finite-volume, step-scaling recursion procedure. We demonstrate that accuracy on the level of a few percent can be achieved by carrying out this matching on a pair of lattices with equal physical spatial volumes but quite different lattice spacings. A fine lattice with inverse lattice spacing and sites and a coarse, , lattice are used together with a heavy quark mass approximately that of the charm quark. This approach is unable to determine the initially expected, four heavy quark parameters: , , , and . This apparent nonuniqueness of these four parameters motivated the analytic result, presented in a companion paper, that this set is redundant and that the restriction is permitted through order and to all orders in where is the heavy quark’s spatial momenta.