Wouldn't the state have to own the hospital to alter the actual care?

In fact, that's how the UK's NHS works. It's like the US's VHA, where the government actually provides health care. It's unlike the US's Medicare, which is "single-payer" because the government pays for everything, but the money goes to private hospitals and doctors who actually provide the health care.

See http://en.wikipedia.org/wiki/Single-payer_health_care and http://en.wikipedia.org/wiki/Socialized_medicine for more information. From the latter:

The original meaning was confined to systems in which the government operates health care facilities and employs health care professionals. This narrower usage would apply to the British National Health Service hospital trusts and health systems that operate in other countries as diverse as Finland, Spain, Israel, and Cuba. The United States' Veterans Health Administration, and the medical departments of the US Army, Navy, and Air Force, would also fall under this narrow definition. When used in this way, the narrow definition permits a clear distinction from single payer health insurance systems, in which the government finances health care but is not involved in care delivery. More recently, American conservative critics of health care reform have attempted to broaden the term by applying it to any publicly funded system. Canada's Medicare system and most of the UK's NHS general practitioner and dental services, which are systems where health care is delivered by private business with partial or total government funding, fit this broader definition, as do the health care systems of most of Western Europe. In the United States, Medicare, Medicaid, and the US military's TRICARE fall under this definition.