In this paper, a numerical method is presented for verifying the existence and uniqueness of solutions to two-point boundary value problems of nonlinear ordinary differential equations. Taking into account every error of numerical computations such as the discretization error and the rounding error, this method also provides mathematically guaranteed error bounds between approximations obtained by numerical computations and the exact solution whose existence is proven by the numerical existence theorem, which is based on the Newton-Kantorovich theorem. Finally, illustrative numerical results are presented for showing the usefulness of the method.