It's the mistake of thinking that when the Church says "right," she means "State obligation." A basic mis-reading.
Just an example: It's like when the Church speaks of a "right to marry" and the gays dive in and say "Us too." Wrong-o. You have to define "marry" as the Church does (Sacramental union, man-woman, eligible to marry, e.g. not already married to somebody else, etc.) and "right" to mean "a natural good to which, in a good society, you have access."
If "right" in this context properly meant "the state must supply it," the State would have to supply spouses to all citizens, I suppose!
Catholics of a generation ago would see medical missions, Catholic hospitals, and fraternal insurance (e.g. Knights of Columbus) --- in other words, voluntary, mutual, and charitable inititives --- as responses to the right to health care as the Church uses the term.
Perhaps therein lies the problem. The only way to absolutely fulfill what they see as a universal "right" (as opposed to privilege) is to have Gov't step in.
The Pope says it is the moral responsibility of nations as opposed to the charity of individuals which implies that the State should step in.
The Vatican Sec'y of State (I assume he's a spokesperson for the Church's position) seems to imply so in this same story
the Secy of State says "Justice requires guaranteed universal access to health care,"
Then goes on to say:
Governments are obligated, therefore, to adopt the proper legislative, administrative and financial measures to provide such care along with other basic conditions that promote good health, such as food security, water and housing.