More seriously, I had not seen this bit by John before and will look forward to reading it (as I look forward to reading Gelman's book at some point). Part of the fun of having John as a colleague is that he thinks really deeply about the philosophical underpinnings of econometrics and statistics.
I do doubt this predictive quality of Gelman's concluding paragraph:
In my experience, any line of argument that relies on John not having read about something is likely to fail.
What I suspect--any readers who know DiNardo can ask him directly--is that he is simply unaware of the modern approach to Bayesian data analysis which is based on modeling and active model checking ("severe testing," to use the phrase of Deborah Mayo). I don't expect that seeing my books would make DiNardo a convert to the Bayesian approach, but it might make him realize that practical Bayesians such as myself are not quite as silly as he might imagine.
Hat tip: Ben Hansen