The president has been sharply criticizing Mr. Tester for days, singling out the Democrat while ignoring Republican opposition that had built to Dr. Jackson’s nomination. Mr. Tester, the ranking Democratic member of the Senate Committee on Veterans Affairs, took the lead in publicly questioning Dr. Jackson’s record, but he had the support of Senator Johnny Isakson of Georgia, the Republican chairman of the committee, who signed a joint statement saying the issues should be investigated.
But Mr. Tester is from a conservative state that in 2016 supported Mr. Trump strongly, giving him 55.6 percent of its votes to 35.4 percent for Hillary Clinton. Even before the flap over Dr. Jackson, Mr. Trump and the Republicans had hoped to use the power of that 20-point margin to defeat Mr. Tester for re-election this fall and defend their narrow 51-seat majority in the Senate.
Mr. Tester released a list of accusations this week against Dr. Jackson alleging loose distribution of prescription drugs, a hostile work environment and drunkenness.
The allegations were raised by more than 20 current and former military personnel who had worked with Dr. Jackson, whose White House medical unit is run by the military, Mr. Tester said, and each of the individual assertions was made by two or more people.
Several of those military officials also described their experiences and concerns about Dr. Jackson to reporters, although they spoke on the condition of anonymity because of their status as members of the military.
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.
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!”
White House aides would not comment on whether the president was referring to anything more than the single episode.
Mr. Tester said when he released his list that he had not sought out the allegations against Dr. Jackson, but that military officers had come to the committee with their concerns. He did not pass judgment on their veracity but said that he had a duty to investigate them and wanted to hear Dr. Jackson’s response.
A spokeswoman for Mr. Isakson, the Republican committee chairman, said on Saturday that he had no comment on the president’s tweets but pointed to the senator’s comment on Thursday when he said: “I think everyone has an obligation if they make a charge to release it. And everyone who is charged has a right to have a day in court to be heard.”
Despite Montana’s conservative tilt, Mr. Tester has appeared to be in good shape to win re-election, and he has expressed no regret and no public concern about the president’s threats. A former chairman of the Democratic Senatorial Campaign Committee, Mr. Tester has raised money at a prodigious rate, and had $6.8 million in his campaign account at the end of the last reporting period.
He has drawn a relatively undistinguished group of Republican challengers. Among them, Matt Rosendale, the state auditor, seems most likely to emerge as the nominee.
As it happens, Mr. Tester caught a significant break last year thanks to Mr. Trump: While national Republicans had settled on Ryan Zinke, then a House member, as the strongest possible challenger for Mr. Tester, Mr. Trump chose him to be interior secretary, removing him from the race.
But Mr. Trump’s easy victory in Mr. Tester’s state means his threats cannot be easily dismissed by Democrats. It is likely that Mr. Trump will expend much of his energy in the midterm elections on deep-red states with Democratic senators, including West Virginia, North Dakota and, evidently, Montana.
National groups are already spending money in the Montana race, with Americans for Prosperity, the organization funded by the conservative billionaires Charles G. and David H. Koch, and the Senate Majority PAC, a Democratic group, beginning new advertising campaigns this month.