As I understand it, the connection is loose.
At one time, the SDA hospitals were legally under the same system of authority as were the SDA congregations, Conferences, etc. Some years back, in order to prevent a lawsuit award against a SDA hospital from bringing down the churches, steps were taken to spin off the clinical parts into a legal organization separate from that of the churches. The result, as I understand it, is that the clinical care facilities may have SDA church leadership on their Boards, but the Church does not control that actions of the clinical section. In a real sense, the clinical sections have a lot of independence.