The overall result, namely, that at the margin medicine has net-negative effects (i.e., after a certain point more medicine equals worse health) has been replicated many times. See for example this meta-study from 2008 about what happens when health workers go on strike [1]:

"[..] mortality either stayed the same or decreased during, and in some cases, after the strike. [No study] found that mortality increased during the weeks of the strikes compared to other time periods."

The authors speculate about the reasons: "The paradoxical finding that physician strikes are associated with reduced mortality may be explained by several factors. Most importantly, elective surgeries are curtailed during strikes. Further, hospitals often re-assign scarce staff and emergency care was available during all of the strikes."

[1] http://www.ncbi.nlm.nih.gov/pubmed/18849101

EDIT: clarity

Also, his idea of increasing taxes will mean that even if his statement were true ("[if] people only bought half as much, they would be just as healthy"), a flat increase in drug tax would at best make this effect true on average.

Due to higher taxation, low-income individuals would have even less access to medicine than today. Keeping people away from drugs with prohibitive taxation means starving entire demographics of this component of health care.

So sure, people who have the funds to take excessive amounts of medicine may be "just as healthy." Others would be considerably worse off.

It's just a bad argument in general...saying that because people spend 30% more when stuff is free (without improved results) means they have twice as much as they need. It doesn't follow, at all.

To be fair, their outcomes were not much worse.