Final answer:
The U.S. government-operated preventive care and hospital system is known as the Veterans Health Administration, an example of socialized medicine specifically for military veterans.
Step-by-step explanation:
The preventive care and hospital system currently operated by the U.S. government is called the Veterans Health Administration. This program is an example of socialized medicine, where the government not only funds but also operates the healthcare facilities and employs the healthcare providers. While other programs such as Medicare and Medicaid provide health care funding for specific demographics, like the elderly and low-income individuals respectively, the Veterans Health Administration is tailored specifically for military veterans as a benefit for their service.