Dr. Jackson called the allegations false and had the support not just of Mr. Trump but also of some former aides to President Barack Obama who said they had never observed the alleged behavior while they worked in the White House. But Dr. Jackson pulled his nomination on Thursday, when it became clear he was unlikely to be confirmed.

“Allegations made by Senator Jon Tester against Admiral/Doctor Ron Jackson are proving false,” Mr. Trump wrote on Twitter on Saturday morning. “The Secret Service is unable to confirm (in fact they deny) any of the phony Democrat charges which have absolutely devastated the wonderful Jackson family. Tester should resign.”

He added: “The great people of Montana will not stand for this kind of slander when talking of a great human being. Admiral Jackson is the kind of man that those in Montana would most respect and admire, and now, for no reason whatsoever, his reputation has been shattered. Not fair, Tester!”

Mr. Tester fired back a few hours later with a statement noting that Mr. Trump had signed eight of the senator’s bills to make the Department of Veterans Affairs more accountable and responsive to veterans. “It’s my duty to make sure Montana veterans get what they need and have earned, and I’ll never stop fighting for them as their senator,” he said.

The White House sought on Friday to refute one of the allegations, that he “got drunk and wrecked a government vehicle,” as Mr. Tester’s list put it. A search of government databases turned up no incident that matched that description, the White House said.

Three cases were found involving Dr. Jackson and a government vehicle, according to the White House: In one, he was rear-ended. In another, a bus sideswiped a mirror on his car. And in a third, a road rage episode, a driver punched through a window on Dr. Jackson’s car.

“Secret Service has just informed me that Senator Jon Tester’s statements on Admiral Jackson are not true,” Mr. Trump wrote on Saturday afternoon. “There were no such findings. A horrible thing that we in D.C. must live with, just like phony Russian Collusion. Tester should lose race in Montana. Very dishonest and sick!”