The dispute was fresh evidence of Trump's willingness to attack any critic and do battle over the most sensitive of matters - and critics' readiness to find fault with his words. On Tuesday, Wilson said Trump made "insensitive" comments to the widow of Army Sgt.
Sergeant La David Johnson was one of four USA military personnel killed almost two weeks ago whose families had not heard from Mr Trump until Tuesday.
Kelly's speech was a rebuke to Wilson, who was in the auto with the family of Johnson when Trump called on Tuesday.
Congresswoman Frederica Wilson said Mr Trump told the widow Sgt Johnson "knew what he signed up for". The slain soldier's aunt and uncle, Richard Johnson and Cowanda Jones-Johnson, who raised him as parents, were in the auto.
'Our president is an S.O.B.' said Katrina Johnson, 42, as she wiped away tears. "I was in the auto and I heard the full conversation".
But that was soon eclipsed by the outrage Kelly expressed over what he believed was Wilson trying to score political points off a tragedy, the officials said.
"She was crying for the whole time", Ms Wilson said of the widow. And he personally absolved Trump - who has frequently struggled showing empathy - of any blame in his call to the family of Sgt. "I don't know", he said of past calls. Relatives of three also said they did not get letters.
Chris Baldridge, the father of Army Cpl.
"I asked them to give me the phone because I wanted to speak with" Trump, Wilson told CNN. But Baldridge said it didn't happen.
White House press secretary Sarah Huckabee Sanders calls on a member of the media during the daily briefing in the Brady Press Briefing Room of the White House, Friday, Oct. 20, 2017. And Trump spokeswoman Lindsay Walters said it was "disgusting" that the news media were casting his "generous and honest gesture" in a negative light.
She said a master sergeant who was in the vehicle declined her request for the phone.
"It stuns me that a member of Congress would have listened in on the conversation - absolutely stuns me", he said.
He later insisted that he "didn't say what that congresswoman said, didn't say it at all". The family member who raised Johnson called Wilson's account of the call "very accurate", and the White House has not refuted that Trump spoke those words.
"Just because the president said "your guy" doesn't mean he doesn't know his name", said Sanders. "He got to talk about what he wanted to talk about and Trump was very receptive and [my dad] is very grateful for the phone call". She also used a dismissive Southwest rancher's term, calling Wilson, who often wears elaborate hats, "all hat and no cattle".
A president's phone calls to bereaved military families are generally not made public, but that's not to say anyone is sworn to secrecy.
What is different is that Mr Trump alone has picked a political fight over who has done better to honour the war dead and their families. "And I would think that every one of them appreciated it", Trump said, noting "those are the toughest calls".
His delay in publicly discussing the men lost in Niger did not appear to be extraordinary, judging from past examples, but his politicisation of the matter is. He went so far Tuesday as to cite the death of Kelly's son to question whether Obama had properly honored the war dead. In response, Kelly told him what General Joseph Dunford, now chairman of the Joint Chiefs of Staff, told him when Robert Kelly was killed. Trump said on Fox News radio. President Barack Obama signed the bill three days before the dedication ceremony. John Kelly then attempted (thanks to the mention of his son by Trump) to defend the president's attempt to console the widow.