Price theory, in particular, and microeconomics theory, in general. Basically, markets approach an equilibrium over time and market equalibria have certain properties, like Pareto Efficiency.

Since that's the only type of domain where we know what "correct" is, I also don't think it can be demonstrated in other domains. Of course, that criticism would apply to any economic system. The point is that the mechanisms are at least self-correcting in theory and that can't be said of other systems.

I like this question. It is often taken for granted in these discussions. Markets are efficient at meeting the various preferences of the populace with the resources available. Specifically, they lead to a state (non-unique) where there is no way to make anyone better off without making someone else worse off (Pareto Efficiency). In a way this is obvious, since all markets do is allow voluntary exchanges, which are win-win (ex-ante). Once all available voluntary exchanges occur, there could be no more win-win exchanges.