A Bayesian approach is developed for analyzing nonlinear structural equation models with nonignorable missing data. The nonignorable missingness mechanism is specified by a logistic regression model. A hybrid algorithm that combines the Gibbs sampler and the Metropolis-Hastings algorithm is used to produce the joint Bayesian estimates of structural parameters, latent variables, parameters in the nonignorable missing model, as well as their standard errors estimates. A goodness-of-fit statistic for assessing the plausibility of the posited nonlinear structural equation model is introduced, and a procedure for computing the Bayes factor for model comparison is developed via path sampling. Results obtained with respect to different missing data models, and different prior inputs are compared via simulation studies. In particular, it is shown that in the presence of nonignorable missing data, results obtained by the proposed method with a nonignorable missing data model are significantly better than those that are obtained under the missing at random assumption. A real example is presented to illustrate the newly developed Bayesian methodologies.