Wilson told several media outlets the president called Johnson at around 4:45 p.m., and that she heard Trump tell the widow "he knew what he was signing up for, but when it happens it hurts anyways." Wilson said the president's call left Johnson in tears.
Johnson's mother, Cowanda Jones-Johnson, said she was also in the car when Trump called her daughter-in-law. She told The Washington Post on Wednesday that Wilson's account of the conversation was accurate. "President Trump did disrespect my son and my daughter and also me and my husband," she said, although she declined to elaborate.
In response to Trump's tweet on Wednesday, Wilson told CNN, "I don't know what kind of proof he could be talking about, but I wasn't the only person in the car. I have proof, too. This man is a sick man. He's cold-hearted and he feels no pity or sympathy for anyone."
The White House on Wednesday did not immediately respond to questions from CNBC about the president's claim to have "proof" that Wilson's account of the call was "totally fabricated."
On Tuesday night, an administration official, speaking on background, told CNBC that "the president's conversations with the families of American heroes who have made the ultimate sacrifice are private."
Trump, however, has made a very public issue of such conversations in recent days, as questions have mounted about why it took the president nearly two weeks to publicly acknowledge the loss of the four American service members killed in Africa.
On Monday, Trump answered a question about his seemingly delayed response to the attack by telling reporters: "I will, at some point, during the period of time, call the parents and the families, because I have done that, traditionally."
Later in the news conference, Trump accused his predecessor, President Barack Obama, of not calling the families of fallen soldiers.
"If you look at President Obama and other presidents, most of them didn't make calls. A lot of them didn't make calls. I like to call when it's appropriate, when I think I am able to do it," Trump said at the impromptu White House press briefing in the Rose Garden.
"So generally I would say that I like to call. I'm going to be calling them — I want a little time to pass — I'm going to be calling them. I have, as you know, since I've been president I have. But in addition I actually wrote letters individually to the soldiers we're talking about and they're going to be going out either today or tomorrow."
The comments drew an immediate backlash from retired military officers and former aides to both Obama and President George W. Bush. A former chairman of the Joint Chiefs of Staff, Gen. Martin Dempsey, was among them.
On Tuesday, Trump admitted that he doesn't really know whether Obama called the families of fallen soldiers, telling Fox News Radio: "Now, as far as other representatives, I don't know. I mean, you could ask General Kelly did he get a call from Obama. You could ask other people, I don't know what Obama's policy was. I write letters and I also call."
According to a statement Tuesday evening from the White House, Trump earlier in the day had called the families of all four soldiers killed in Niger.
"He offered condolences on behalf of a grateful nation and assured them their family's extraordinary sacrifice to the country will never be forgotten," White House press secretary Sarah Huckabee Sanders said in the statement.