Trade and Health Care

Over at Eyes on Trade, Todd Tucker says: "a move to universal health care could be challenged as a limitation on market access for health insurance companies."  Upon reading this, my first thought was to look at the relevant part of the U.S. GATS Schedule to see what commitments had been made.  However, when I did so my eyes immediately glazed over.  There are lots of "nones" and "unbounds" and "horizontals."  I assume I could figure it out if I really tried, but I couldn't get up the energy just now.  But an interesting thought did occur to me:  As long as private insurance companies are still involved, couldn't universal health care expand market access by increasing the number of insured people (that is, the number of customers)?