No, they do not have the authority to verify whether or not what you're taking through and claiming to be medicine is actually medicine. TSA does however have the authority to verify that the item you're claiming to be medicine is not WEI. That's as far as it goes.

There is a difference. Suppose I'm brining 8 medications through the checkpoint. TSA can X-ray the bottles to make sure there's no WEI hidden in them. They can swab the outside of the bottles to ensure there's no WEI in them. They may even be able to ask that the medication be transferred to another bottle or baggie if a swab for WEI comes up positive on the bottle.

But TSA at no time has the authority to take the medication from me and verify that it actually is medication (i.e. check its formulation, call the pharmacy and verify, etc). For all TSA knows it could be Smarties, blood pressure medication, or little pieces of crack cocaine in the bottles-- but they have no authority (and no business) taking the examination anywhere outside the narrow, limited scope of searching for WEI...

To support otherwise starts bringing us toward the slippery slope of Fourth Amendment violations and illegal government-run dragnet operations.