I understand that RMSE is a useful summary measure for assessing predictive power of regression model with continuous dependent Y. Is RMSE equally applicable for logistic regression or is there a better summary measure of predictive power?
Logistic Regression is a model estimated via Maximum Likelihood, so you won't have many of those beautiful measures from OLS regression. But don't worry there are some cool stuff you can use. Particularly, for measuring predictive power you have ROC curves and two important measures: Sensitivity and specificity. These are the standard values to analyze the predictive capabilities in logistic regression:
In order to measure goodness of fit, the most reliable measure for comparing models is the Likelihood ratio test. Most works will use this value when comparing two models. The sad part of likelihood ratio test is that it will only be valid for nested models and exactly equal data sets (i.e. you cannot compare a model for men against another for women). Also if you estimated a model with Restricted Maximum Likelihood, the likelihood ratio may not be valid. In case you want to compare non nested models, you use AIC or BIC and its respective ratio test. Wald test is a more general approach although it is only reliable with larger samples.