On Tuesday, Mr. Trump dragged his chief of staff, John F. Kelly, into the dispute. He told reporters that Mr. Obama had not called Mr. Kelly, a former Marine Corps general, when his son Second Lt. Robert Kelly was killed in action in 2010 in Afghanistan. Mr. Kelly, who has long been reluctant to talk about the loss of his son, did not comment on the issue.
But the White House presented Mr. Kelly as a character witness on Wednesday, noting that he was present for Mr. Trumpâs call on Tuesday afternoon to Sergeant Johnsonâs wife, Myeshia Johnson, and viewed it as a respectful expression of presidential sympathies.
âHe thought that the president did the best job he could under those circumstances to offer condolences on behalf of the country,â said Sarah Huckabee Sanders, the White House press secretary. She said that Mr. Kelly is âdisgusted by the way this has been politicized, and that the focus has become on the process and not the fact that American lives were lost.â
It was Mr. Trump, however, who first put a spotlight on politics and process by comparing his practices with those of Mr. Obama and other presidents. Mr. Obama did, in fact, call or meet with the families of multiple fallen soldiers, though he sent letters to many others. Mr. Trump said he planned to call as many families of fallen soldiers as was âappropriate.â
His call to Ms. Johnson came as she and her two young children were in a limousine at Miami International Airport awaiting a plane carrying the remains of Sergeant Johnson. Mr. Trump spoke for three to five minutes, Ms. Wilson said.
âWhen she got off the phone, she said, âHe didnât even know his name. He kept calling him, âYour guy,ââ Ms. Wilson said of Ms. Johnson. âHe was calling the fallen soldier, âYour guy.â And he never said his name because he did not know his name. So he kept saying, âYour guy. Your guy. Your guy.â And that was devastating to her.â
Mr. Trump flatly dismissed Ms. Wilsonâs account and suggested he would produce evidence to discredit it.
âDemocrat Congresswoman totally fabricated what I said to the wife of a soldier who died in action (and I have proof). Sad!â he wrote in an early-morning Twitter post. He repeated his denial hours later, before a White House meeting with senators. âI didnât say what that congresswoman said,â the president said. âDidnât say it at all, she knows it.â
Ms. Wilson quickly fired back on Twitter. âI still stand by my account of the call b/t @realDonaldTrump and Myeshia Johnson. That is her name, Mr. Trump. Not âthe womanâ or âthe wife,ââ she wrote in a post.
Sergeant Johnsonâs mother, Cowanda Jones-Johnson, backed the congresswomanâs version. âYes, he did state that comment,â Ms. Jones-Johnson said, via message on Facebook, of Mr. Trumpâs remark that her son âknew what he signed up for.â
By midafternoon, the White House was no longer disputing Ms. Wilsonâs account of Mr. Trumpâs choice of words. Ms. Sanders said the White House did not tape the call. But she said Ms. Wilson had willfully mischaracterized the spirit of the conversation.
âThis is a president who loves our country very much, who has the greatest level of respect for men and women in uniform, and wanted to call and offer condolences to the family,â Ms. Sanders said. âTo try to create something from that, that the congresswoman is doing, is frankly appalling and disgusting.â
The dispute over Mr. Trumpâs condolence call topped several contentious issues that marked yet another rancorous day at the White House.
The president kept up his feud with the National Football League over players who take a knee in protest during the playing of the national anthem. And he revived his unproven charges that the former F.B.I. director, James B. Comey, had lied, leaked information and protected Hillary Clinton, Mr. Trumpâs opponent in last yearâs presidential election.
But the way Mr. Trump has handled grieving military families loomed over all, and thrust the sensitive issue of how presidents deal with the casualties of war to center stage. His reference to Mr. Obamaâs lack of calls also drew furious responses from the former presidentâs aides and expressions of discomfort from former military commanders.
The feud with Sergeant Johnsonâs family was reminiscent of a public fight Mr. Trump began with the parents of a Muslim American soldier, Humayun Khan, who was killed in 2004 in Iraq. The soldierâs parents, Khizr and Ghazala Khan, appeared at the Democratic National Convention in 2016, where Mr. Khan criticized Mr. Trump.
Mr. Trumpâs charged relationship with Gold Star families â those who have lost relatives in war â took another turn with the White Houseâs disclosure on Wednesday that it had sent a check for $25,000 to the family of Sgt. Dillon Baldridge, who was shot to death by an Afghan police officer, along with two other American soldiers, in June.
Mr. Trump had promised the check to Sergeant Baldridgeâs father, Chris, in a phone call a few weeks after his sonâs death, according to The Washington Post. But the president did not send the money until the newspaper inquired about it on Wednesday.
âFor somebody to tell me they were going to give me something and then not come through, it feels like kicking me when Iâm down,â Mr. Baldridge said Wednesday.
Some experts sympathized with the challenge Mr. Trump faced in placing condolence calls.
âItâs always been difficult for presidents,â said Peter D. Feaver, an expert in civilian-military relations at Duke University, âbut in some ways, itâs become more difficult as the number of casualties dwindled, so each one can be individualized to a much greater extent.â
Other calls Mr. Trump has made to families have been well received. The president called Eddie Lee, the father of First Lt. Weston C. Lee, who was killed in April by a roadside bomb in Iraq, and told him, âI bet he never gave you a minuteâs trouble as a child.â
âItâs true,â Mr. Lee said, chuckling, âhe didnât.â
âThe president was just so nice and caring, you could hear it in his voice, you could tell what a caring family man he is,â said Mr. Lee, who volunteered, âI voted for Trump and Iâd vote for him again.â
But the presidentâs call to Sergeant Johnsonâs widow illustrated the pitfalls to his improvisational approach, according to other experts. Kori Schake, a research fellow at the Hoover Institution of Stanford University who specializes in civilian-military relations, said the account of Mr. Trumpâs call suggested he did not follow the âtime-honored ritualsâ of such calls.
âMy guess is that he thought he was showing respect for the toughness and patriotism of people who sacrifice for something bigger than themselves, and just did it clumsily,â Ms. Schake said.
An earlier version of this article referred imprecisely to Sgt. La David T. Johnson. He was serving in Africa with an Army Special Forces unit; he was not a Green Beret.