In a word, no. The reason is that if the bayesian does a really crappy job of specifying their prior distribution, the posterior will be completely inaccurate.

In a word, no. The reason is that if the bayesian does a really crappy job of specifying their prior distribution, the posterior will be completely inaccurate. On the other hand, if the bayesian does a good job, getting intervals as you describe will work much better (e.g. be smaller) than confidence intervals.