Lt. Randall J. Clark, Mr. McCloughan’s onetime platoon leader, had inquired after the battle about awarding the medic the Distinguished Service Cross, but he was given a bronze star for valor instead. It was not until 2009, when Mr. McCloughan’s uncle secured him a meeting with Representative Fred Upton, Republican of Michigan, that the topic was revisited. With the help of Mr. Clark, Mr. Upton and Michigan’s congressional delegation, the case eventually got the attention of Ashton B. Carter, President Barack Obama’s defense secretary, who recommended last year that Mr. McCloughan be awarded the Medal of Honor.

But because the award must be given within five years of the actions it recognizes, Congress had to vote to grant special permission. By the time it had done so, the Obama administration ran out of time to invite Mr. McCloughan to the White House.

Mr. McCloughan, who was known as Doc to members of his platoon, was drafted into the Army shortly after he finished college, in 1968. By March 1969, he was assigned to a base in South Vietnam and two months later found himself part of a dwindling company tasked with securing a transportation route near Tam Kỳ.

When they flew by helicopter into the area, the Americans almost immediately came under fire. Though they did not realize it yet, more than 2,000 North Vietnamese and Vietcong soldiers had their position surrounded. With his company under fire and in retreat, Mr. McCloughan hoisted his first downed soldier onto his shoulders and evacuated him to safety.

By late afternoon, he again sprinted forward, weaponless, to pull two more stranded men back to a trench (likely dug by French troops in the 1950s). This time, he was hit. Mr. McCloughan looked down and saw he was covered with blood. Shrapnel from a rocket-propelled grenade had hit his head and body.

Over the two-day battle, Mr. McCloughan voluntarily risked his life seven more times, according to an account of the battle released by the White House, returning fire and, in one case, taking out an enemy rocket-propelled grenade position. When a superior ordered him to get in a medevac helicopter alongside one of the men he was treating, Mr. McCloughan refused.

“As Jim now says,” Mr. Trump said, “‘I would have rather died on the battlefield than know that men died because they did not have a medic.’”