We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In our previous work for the wave equation, two pairs of order-preserving interpolation operators were needed when imposing the interface conditions weakly by a penalty technique. Here, we only use one pair in the ghost point method. In numerical experiments, we demonstrate that the convergence rate is optimal and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as that given by the usual Courant-Friedrichs-Lewy condition.