Critical properties in the dynamical charge correlation function for the one-dimensional Mott insulator are studied. By properly taking account of the final-state interaction between the charge and spin degrees of freedom, we find that the edge singularity in the charge correlation function is governed by massless spinon excitations, although it is naively expected that the spinons do not directly contribute to the charge excitation over the Hubbard gap. We obtain the momentum-dependent anomalous critical exponent by applying the finite-size scaling analysis to the Bethe ansatz solution of the half-filled Hubbard model.