A nonlinear theory for the calculation of density and polarization of dipolar solvents near charged surfaces is developed by using a density functional approach. The theory is based on a weighted density approximation for the isotropic part of the density and the anisotropic (or dipolar) part is calculated by using a perturbative approach. The theory, however, retains the full nonlinear dependence on the surface charge density or the external field. Explicit numerical results are obtained for different values of the external field. It is found that the number density of the solvent near the surface increases with increasing surface charge density indicating the presence of electrostriction. The polarization is found to increase nonlinearly with the external field and exhibits the presence of dielectric saturation at high field strength. The predictions of the present nonlinear theory are compared with the results of Monte Carlo simulations and a good agreement is found for both the density and polarization near charged surfaces. The present simulations clearly reveal a nonlinear behavior of dipolar molecules in presence of the charged surfaces. Also, the nonlinear effects are found to be especially important for interfacial molecules, in agreement with the predictions of the analytical theory presented here.