We develop a semiparametric Bayesian inference framework, encompassing estimation and model comparison procedures, for models characterized by conditional moment conditions. Our techniques utilize and complete the Bayesian exponentially tilted empirical likelihood (BETEL) framework developed in Chib, Shin and Simoni (2018) for unconditional moment condition models. The starting point is a conversion of the conditional moments into a sequence of unconditional moments by using a vector of approximating functions (such as tensor splines based on the splines of each conditioning variable) with dimension that is increasing with the sample size. We establish that the BETEL posterior distribution satisfies the Bernstein-von Mises theorem, subject to a rate condition on the number of approximating functions. We also develop an approach based on marginal likelihoods and posterior odds ratios for comparing nested and non-nested and misspecified conditional moment restricted models and establish the model selection consistency of our procedure. Illustrative examples are provided.