As the second-most senior person at the Department of Veterans Affairs, one of the government’s largest and most historically troubled agencies, Mr. Byrne had broad authority over a number of sensitive issues and important initiatives as the department moves to direct thousands of veterans to health care providers outside the agency’s legacy network.

The agency’s $18 billion attempt to modernize its health care records has been troubled with its communications with the Defense Department system, and with maintaining enough staff for an ambitious debut deadline next month. As the agency moves to get more health care for veterans in their community under the new plan, it has struggled to increase its network of doctors to meet the popular demand.

An initial task force plan to combat veterans suicide — a key policy initiative — has not been met with satisfaction at the White House, three people familiar with the plan said, in part because it is viewed as focusing too narrowly on reducing firearms, which are a leading method of suicide. Mr. Byrne alluded to the emphasis on veterans suicide as a Trump administration priority at his confirmation hearings.

Last fall, a senior policy adviser on female veterans issues for the House said she was assaulted at the V.A. Medical Center in Washington, a complaint that was sent to the inspector general, in part at the behest of Representative Mark Takano, Democrat of California and chairman of the Veterans Affairs Committee.

The inspector general later declined to bring any charges, in part because video cameras at the hospital did not capture the episode. In an unusual rebuke of a committee chairman, Mr. Wilkie wrote to Mr. Takano that “the unsubstantiated claims raised by you and your staff could deter our veterans from seeking the care they need and deserve.”