We consider stochastic wave equations in spatial dimensions $d \geq 4$. We assume that the driving noise is given by a Gaussian noise that is white in time and has some spatial correlation. When the spatial correlation is given by the Riesz kernel, we also establish that the spatial integral of the solution with proper normalization converges to the standard normal distribution under the Wasserstein distance. The convergence is obtained by first constructing the approximation sequence to the solution and then applying Malliavin-Stein's method to the normalized spatial integral of the sequence. The corresponding functional central limit theorem is presented as well.