(It seems... implausible that this will actually happen, but then no-one has ever gone broke betting on Trump doing bloody stupid things, so there's that...)
Drug companies simply won't test for safety (or lie about the results). Any 3rd party will be seen as worse than worthless. They'll either be funded by the competition to undermine trust in competitors or they'll not have the funding needed to advertise their results (they'll be indistinguishable from cranks).
And further, No free market mechanism will actually have enforcement mechanisms. Which means unsafe medicine will stay on the market indefinitely.
Testing for safety is a costly endeavor that pharmaceutical companies will pretty happily ditch.
We know what the US without the FDA looked like. The FDA wasn't always there. In that bygone era coke was sold as a health drink and milk was blue. [2]
Oh, and realize we have emails from executives about hiding the evidence that vioxx causes heart attacks, before the drug was approved.
If you have doubts, remember that J&J convinced the FDA that it will handle it's own testing for safety in regards to asbestos in baby powder. That was in the 70's - it took 50 years for J&J's crimes against humanity to come to air.