"I want to ask you a last question before you leave," Matthews told Trump, the Republican presidential front-runner.
"You can leave, but I'd like you to stay," the "Hardball" host continued. "Is Donald Trump honest when he says that Barack Obama isn't a legitimate president? This is a good question."
Matthews, a former Democratic congressional aide, and Trump then talked over each other.
"I should not tell you this, but I do watch you a lot," Trump said. "So I knew you were going to ask that question. And you know what I'll say: I don't talk about that anymore."
Trump was once one of the most prominent people to question the authenticity of Obama's birth certificate. Conspiracy theorists long raised doubts about the president's birthplace, suggesting that he was actually born in Africa instead of Hawaii.
Obama's long-form birth certificate was eventually released in 2011, but Trump doesn't admit defeat. Instead, as Trump has repeated more than once during his presidential campaign, he simply refuses to answer the question.
"You know the problem talking about that question?" he told Matthews after the Tuesday debate, which was hosted by CNN. "Here's the story. I don't answer because you know what? If I do answer, that's all people want to talk about."
Trump and Matthews continued to squabble.
"You're going to take the oath of office and [believe] the president leaving the office is illegitimate?" Matthews asked.
"I don't answer that question because once I answer the question, they don't want to talk about the economy, they don't want to talk about all the other" issues, Trump replied.
"We Catholics believe in confession: You say you were wrong and you move on. You really believe this guy is an illegitimate president?" Matthews pressed.
"I don't want to answer the question," the mogul maintained. "I don't want to answer the question."
The interview then awkwardly ended.