WASHINGTON — President Trump’s dismissal of David J. Shulkin, the secretary of veterans affairs — and the nomination of a Navy doctor with no known policy views to take his place — has brought renewed focus to an increasingly contentious debate over whether to give veterans the option of using the benefits they earned through military service to see private doctors rather than going to government hospitals and clinics.

The issue, which has pitted almost every major veterans group against Concerned Veterans for America, an advocacy group funded by the billionaire conservative brothers Charles G. and David H. Koch, and its allies, has been at the center of months of intrigue at the sprawling Department of Veterans Affairs, which is charged with caring for the United States’ 20 million veterans.

But Mr. Shulkin’s departure and the abrupt elevation of Dr. Ronny L. Jackson, the White House physician, to the department’s top job on Wednesday have raised new fears among Democrats and groups like the Veterans of Foreign Wars and the American Legion. They worry that the Trump administration will push for a major change in veterans’ health care that they have bitterly opposed.

The groups say the end result would be disastrous, effectively bleeding to death a network of 1,700 hospitals and clinics that has taken decades to build.