Matched-field acoustic source localization is a challenging task when environmental properties of the oceanic waveguide are not precisely known. Errors in the assumed environment (mismatch) can cause severe degradations in localization performance. This paper develops a Bayesian approach to improve robustness to environmental mismatch by considering the waveguide Green's function to be an uncertain random vector whose probability density accounts for environmental uncertainty. The posterior probability density is integrated over the Green's function probability density to obtain a joint marginal probability distribution for source range and depth, accounting for environmental uncertainty and quantifying localization uncertainty. Because brute-force integration in high dimensions can be costly, an efficient method is developed in which the multi-dimensional Green's function integration is approximated by one-dimensional integration over a suitably defined correlation measure. An approach to approximate the Green's function covariance matrix, which represents the environmental mismatch, is developed based on modal analysis. Examples are presented to illustrate the method and Monte-Carlo simulations are carried out to evaluate its performance relative to other methods. The proposed method gives efficient, reliable source localization and uncertainties with improved robustness toward environmental mismatch.