A White House official said Mr Trump's conversations with the families of dead servicemen were private.
Mr Trump later told reporters: "I did not say what she [Ms Wilson] said... I had a very nice conversation."
When asked about what "proof" he could offer, Mr Trump said: "Let her make her statement again, then you will find out."
Ms Johnson responded to Mr Trump's denial by tweeting: "I still stand by my account of the call b/t @realDonaldTrump and Myeshia Johnson. That is her name, Mr Trump. Not "the woman" or "the wife".
By Anthony Zurcher, senior North America reporter, BBC News
In US politics, nothing is off-limits any more.
After (inaccurately) swiping at his predecessors for not calling the family members of US soldiers killed in combat, Mr Trump is on the defensive over allegations he mishandled a call with a grieving widow.
The accuser is a partisan Democratic congresswoman and the president, not surprisingly, is pushing back hard. This controversy is spiralling towards the gutter.
Mr Trump made this bed, however. He was quick to cite the slain son of chief of staff John Kelly to justify his contention that Barack Obama didn't always make phone calls. Then there were the disparaging comments candidate Mr Trump made last summer about the parents of a Muslim-American soldier killed in Iraq.
The more this story drags on - and it will drag on - the more damage it could do to a president who wraps himself in the symbols of patriotism and the military, but is in danger of being viewed by the public as lacking empathy when it counts most.
An important presidential role is consoler-in-chief during times of tragedy. Successful politicians learn early that they need a human touch.
It's a job Mr Trump, the anti-politician, has little experience doing - and it shows.
How did this row begin?
Mr Trump has been on the defensive over the deaths in Niger since a reporter asked him at the White House on Monday why he had still not called the families.
He provoked fury by falsely claiming that his predecessor, Barack Obama, and other former US presidents had not called the relatives of dead service members.
Mr Trump also said he had written letters to the families of the four servicemen killed in Niger and planned to call them soon. The White House later said the president had spoken to the families but it did not say when.