Intractability of NP-complete problems as a principle of physics?
I don't think that the fact that $P\neq NP$ is an asymptotic statement is an automatic "dealbreaker". One can make concrete conjectures that are consistent with our knowledge but stronger than P vs NP such as "It takes at least $2^{n/10}$ steps to find a satisfying assignment for a random n variable 10SAT formula" (with "random" being e.g., the planted model of Achlioptas Coja-Oghlan, this is just an example - I don't know what are reasonable concrete numbers offhand). 

Such conjecture can result in a refutable prediction that any natural system that will try to solve this will fail (e.g., get stuck in a local minima), something you can verify with experiments. Indeed, I'm not an expert on this but to my knowledge, as Joe Fitzsimons mentioned, such predictions had been confirmed with Adiabatic computing. (Scott Aaronson also had some entertaining experiments with soap bubbles.) 

Of course you can also see some "empirical evidence" for $P \neq NP$  in the fact that people have been trying to solve optimization problems, cryptoanalyze encryptions, etc.. and haven't been successful so far...