I disagree -- are you referring to chapter 6 of BDA? In that chapter he spells out good ways of addressing the issue: the Bayesian analogs of classical hypothesis testing statistics. Most importantly, though Gelman doesn't use this language, is the idea of devising test statistics that would falsify your model and then using bootstrapping methods to compare those test statistics on the posterior distribution to the test statistics on the data. In my own view, this is a shining success of Bayesian methods over frequentist methods. Bayesian analysis might give you intractable posterior distributions, and the test statistics that matter for falsifiability are hardly ever going to have convenient forms like F-distributions, t-distributions, or Chi-squared distributions, as are naively advertised in the classical approaches. But computational methods like Metropolis-Hastings/Gibbs sampling and other advances in MCMC still let you bootstrap the test statistic even when its distribution is impossibly complicated. I think this advantage of Bayesian methods deserves to be more widely understood. The other notions mentioned in chapter 6 of BDA are graphical data analysis and measures for model expansion / predictive accuracy.

In the paper, it seemed that the part Gelman refused to address was the way in which the addition of model checking / going back to the drawing board ruined the logical coherence of the more usual inductive Bayesian arguments. I agree that he copped out here and didn't attempt to address the underlying philosophical problem -- all he did was point out that each of the other major alternatives has basically the same coherence problem, including inductive Bayes.