During an appearance on Fox News on Wednesday, Mr Giuliani was asked "how damaging would it be for a Biden administration to criminally prosecute a former president?"
"Well, we would become a banana republic", he said in response. "And that's where we're headed."
The comments came as part of a segment titled, "How Kamala Harris would abuse the Department of Justice (DOJ)". Sen Harris, a California senator, was chosen as Mr Biden's running mate on Tuesday.
Sen Harris previously said during her failed presidential run that Mr Trump would be prosecuted for alleged obstruction of justice if she were elected president.
Under her leadership the Department of Justice “would have no choice” but to pursue criminal charges against Mr Trump due to revelations in Robert Mueller’s Russia report, the California senator said in 2019.
Mr Biden said in an interview last week that he would not stand in the way of a future Justice Department pursuing criminal charges.
"I will not interfere with the Justice Department's judgment of whether or not they think they should pursue the prosecution of anyone that they think has violated the law," Mr Biden told NPR's Lulu Garcia-Navarro.
"In terms of saying, 'I think the president violated the law. I think the president did this, therefore, go on and prosecute him' — I will not do that," he said.
Mr Giuliani went on to call Sen Harris a "mean woman" and a "terrible prosecutor" echoing the sentiments of the president, who called the senator "nasty" directly after her announcement as Democratic running mate on Tuesday.
During their first appearance as running mates on Wednesday, Mr Biden hit back Mr Trump's attacks on Sen Harris.
“It's no surprise,” Mr Biden said. “Whining is what Donald Trump does best more than any president in American history. Is anyone surprised Donald Trump has a problem with a strong woman?”
Sen Harris broadly condemned Mr Trump and his administration but didn't address his specific remarks about her.