We propose a debiased Bayesian inference method for the average treatment effect (ATE) for binary outcomes under conditional unconfoundedness. Under unconfoundedness, the ATE can be written as a population average of conditional means. We use a nonparametric Bayesian approach for these conditional mean functions, which builds on a data-dependent prior. We propose a novel correction term that builds on residuals weighted by the inverse propensity score. Due to this correction, our semiparametric Bayesian approach resembles the efficient influence function of frequentist ATE estimation. In fact, we show asymptotic equivalence of our debiased Bayesian estimator and efficient frequentist estimators by establishing a version of the Bernstein-von Mises theorem. In particular, we show that Bayesian credible sets form confidence intervals in the frequentist sense with asymptotically accurate coverage probability. Our debiased Bayesian inference results require smoothness conditions only of a “double-robust” form that allows the smoothness of the regression function to be compensated by the smoothness of the propensity score and vice versa. In simulations, we find that our debiasing correction leads to accurate coverage of confidence intervals. We illustrate our method in an application to the National Supported Work Demonstration.