The two have been sparring on Twitter and at the earlier debate, Pataki, who now lives in Garrison, said Trump will not be the Republican nominee.
"He is unfit to be president and to be the GOP nominee," Pataki said.
Pataki brought up Trump's time in Atlantic City, saying he invested in four casinos, all of which went bankrupt, causing 5,000 people to lose their jobs.
"He didn't lose anything," Pataki said. "He will do for America what he did for Atlantic City."
At the evening debate, Trump fired back saying that Pataki was a failed governor.
"He wouldn't be elected dog catcher," Trump said.
Trump said he made a tremendous amount of money in Atlantic City and got out before the city collapsed.
At the debate, Trump sparred with Sen. Rand Paul saying he shouldn't be on the stage and said he could go after his appearance.
"There's plenty of subject matter right here," Trump said.
Trump took a shot at Gov. Scott Walker, saying Wisconsin has lost $2.2 billion since he took office and refused to apologize to Columba Bush, Gov. Jeb Bush's wife, for comments he made about how people should speak English in America.
Trump also sparred with Carly Fiorina over their business records and comments Trump had made about her appearance to "Rolling Stone," specifically her face.
"I think women all over this country heard very clearly what Mr. Trump said," Fiorina said.
"I think she's got a beautiful face and is a beautiful women," Trump responded.
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