Ms Burley asked General Jack Keane if the US President’s language had been appropriate during a phone call with the widow of a United States soldier killed in action.
Donald Trump has denied telling the widow of Sgt La David Johnson that “he knew what he signed up for”.
General Keane refused to reply to the Sky News presenter and, at one point, sat silent while the host repeatedly asked him to clarify his position.
After the former US Army Vice Chief of Staff said he “suffered for the widow’s loss”, Ms Burley asked why Mr Trump had not said that during the phone call.
She said: “Why didn’t the President say that, I wonder?
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“We want to know exactly what you think about this, sir.”
As the interview became more heated and General Keane said he was unwilling to continue with the topic, Ms Burley challenged him further.
She said: “You’re going to walk out if we continue to talk about this conversation, is that what you’re saying?”
“I’m asking you a follow up question. You said you would never challenge a war widow, I’m saying to you that your President did, is that acceptable?
“What is your answer? Remind me please.
“Do you appreciate how offensive this must be to war widows, not just of the one that we have been talking about, but more generally.
“When someone in your position feels it is inappropriate to answer these sort of questions.”
General Keane did not hold back as he told the news presenter it was time to discuss something else.
He said: “Oh come on, can we stop this conversation and get on with something else please.
“I’ve already given you my answer. I’m done with the conversation.
“I’m done with the conversation, make up your mind.
“I think I’ve answered your questions, alright.”
The US President said it was not true he made the comments to Myeshia Johnson in a phone call to express his condolences over the death of her husband.
Mr Trump reportedly said: “Well I guess he knew what he signed up for but I guess it still hurt.”
His remarks were recounted by Democrat Representative Frederica Wilson who said that Mr Trump was must have a “brain disorder”.
The President tweeted that Miss Wilson “fabricated” the quotes to score political points.