I think the group that most reliably and effectively funds global health -- at least in terms of total $ -- would be the United Nations, or perhaps the Catholic Church, or otherwise one national government or another.

If you exclude "nations" then it does look to be the Church: "The Church operates more than 140,000 schools, 10,000 orphanages, 5,000 hospitals and some 16,000 other health clinics". Caritas, the relevant charitable umbrella organization, gives $2-4b per year on its own, and that's not including the many, many operations run by religious orders not under that umbrella, or by the hundreds of thousands of parishes around the world (most of which operate charitable operations of their own).

And yet, rationalists are totally happy criticizing the Catholic Church -- not that I'm complaining, but it seems a bit hypocritical.