The primary objective of this research is to develop a novel high-order symmetric and energy-preserving method for solving two-dimensional nonlinear wave equations. Initially, the nonlinear wave equation is reformulated as an abstract Hamiltonian ordinary differential equation (ODE) system with separable energy in an appropriate infinite-dimensional function space. Subsequently, an energy-preserving and symmetric continuous-stage Runge-Kutta-Nyström time-stepping scheme is derived. After approximating the spatial differential operator using the two-dimensional Fourier pseudo-spectral method, we derive an energy-preserving fully discrete scheme. A rigorous error analysis demonstrates that the proposed method can achieve at least fourth-order accuracy in time. Finally, numerical examples are provided to validate the accuracy, efficiency, and long-term energy conservation of the method.