Wilson has accused Trump of offending the family with his comments and Kelly said it was inappropriate for the lawmaker to listen in on the call.
"There's no ideal way to make that phone call", he said in his first public comments on the controversy over the call, which Johnson's mother and a Florida lawmaker who knew him said was disrespectful.
"He said to me, 'What do I say?' 'I said to him, 'Sir, there's nothing you can do to lighten the burden on these families, '" he said.
Sanders said Wilson made the comments referenced by Kelly and it was "witnessed by many people that were there". The Democratic congresswoman had criticized Trump's recent condolence call to the widow of Sgt. La David Johnson, one of four soldier killed in Niger earlier this month. At one point, he scolded Rep. Wilson for supposedly bragging about getting securing funding for an Federal Bureau of Investigation field office in 2015.
"Immediately I went into attack mode", Wilson said, noting that others had told her the process could take as long as a year.
The White House did not directly respond to the video that contradicts the chief of staff's claims, and instead repeated Kelly's criticism of Wilson as "an empty barrel".
"I was stunned when I came to work yesterday morning and brokenhearted at what I saw a member of Congress doing", Kelly said.
"That was just simply to say I don't believe President Obama called".
Seven policemen killed in Pakistan blast
Earlier in June, 14 people ─ including seven policemen ─ were killed in a powerful blast in Quetta's Gulistan Road area. Also Wednesday, a roadside bombing near a tribal area along the border with Afghanistan killed two troops.
A Johnson family member, who also heard the conversation, backed up Wilson's account. In response, Kelly told him what General Joseph Dunford, now chairman of the Joint Chiefs of Staff, told him when Robert Kelly was killed. "He knew what the possibilities were because we're at war", Kelly says he had been told by a friend and colleague. The family of Army Sergeant Jonathon Hunter, who was killed in an August suicide bombing in Afghanistan, was told to expect a call from Trump. But a video of her remarks at the dedication shows she never took credit for getting the government to come up with the money for the project.
Further, Wilson claimed that one of Kelly's terms for her - an empty barrel - was "racist", even though she also said she hadn't heard of an empty barrel before.
But the president, he said, insisted that he thought it was the right thing to do. Those calls, he said as he fought back tears, were the most important.
Kelly said he was appalled that Wilson actually overheard what the president said to the widow.
Brianna Keilar was struck by Huckabee Sanders' "absurd statement" about the media continuing to cover this, while former retired Rear Admiral and State Department spokesman John Kirby agreed that of course Kelly "should be questioned" and scrutinized by the press.
Kelly said it was this message that Trump was trying to convey to Johnson's widow in his own way.
But Trump wanted to make the calls, and asked Kelly for advice on what to say.