At EconLog Bryan argues that one reason constitutions matter is that the content of the constitution has an affect on what people will endorse. He cites a poll showing that more people say they like free speech if the language of the question ties it to the Constitution.
I bet examples like this would be easy to multiply. I suspect, for example, that the Supreme Court's rulings against regulation during the Lochner era not only restrained majority excesses; they also probably reduced the majority's support for regulation. No wonder political activists spend so much time in seemingly fruitless quarrels about “what the Constitution really means.” While many people seem to think that the Constitution always favors whatever policy they prefer, there are actually quite a few people who prefer whatever policy they think the Constitution favors.
Is this empirical evidence for Rawls's claim that constitutions create a basis for public deliberation and justification? Well, maybe. Here is, at least, a possible explanation/justification for the phenomenon Bryan observes.
The fact that a principle appears in the constitution implies that it was the outcome of an earlier political consensus. A citizen can reasonbably assume that there was some good reason why consensus settled on this principle, and so the principle has some support simply in virtue of being embedded in the constitution. In the absence of evidence that a constitutional principle is causing a problem, the rationally ignorant citizen is correct, ceterus paribus, to give more weight to a priciple appearing in the constitution than a principle not appearing in the constitution. Since rationally ignorant folk won't know what principles are in the constitution, a poll question that tells them that a principle is, while asking them if they agree with it, is likely to enjoy a more positive response.
This is just a specific example of the general principle that it is epistemically rational for cognitively limited agents to respect the cognitive division of labor and defer to epistemic authority. If you're shooting in the dark for epistemic authorities (identifying them is itself an epistemic problem) framers of constitutions and supreme court judges are good guesses.