Doubly robust estimators are typically constructed by combining outcome regression and propensity score models to satisfy moment restrictions that ensure consistent estimation of causal quantities provided at least one of the component models is correctly specified. Standard Bayesian methods are difficult to apply because restricted moment models do not imply fully specified likelihood functions. This paper proposes a Bayesian bootstrap approach to derive approximate posterior predictive distributions that are doubly robust for estimation of causal quantities. Simulations show that the approach performs well under various sources of misspecification of the outcome regression or propensity score models. The estimator is applied in a case study of the effect of area deprivation on the incidence of child pedestrian casualties in British cities.