“Some government lawyer would be nervous to do what I do,” he said in an interview on Friday. “I’m a private lawyer, I represent my client, and I’m going to prove it to you that he’s innocent. Whether you like it or not, somehow I’m going to eventually get you to cover it.”

It’s far from clear that Mr. Giuliani’s argument — that a whistle-blower complaint filed by an intelligence official against Mr. Trump will “turn out to be even stupider than Papadopoulos” — will prevail. Mr. Trump, according to multiple sources, pressured the president of Ukraine to investigate Mr. Biden’s son, Hunter, and urged him to work with Mr. Giuliani on such an inquiry. Mr. Trump and Mr. Giuliani have denied discussing the former mayor’s contacts with Ukrainian officials.

Mr. Giuliani described his 28 minutes of cross talk with Mr. Cuomo on Thursday night as “pretty rough.” During the exchange, Mr. Giuliani said he had no idea whether Mr. Trump spoke with the Ukrainian president about Mr. Biden, or Mr. Trump’s former campaign chairman, Paul Manafort. But if he did, Mr. Giuliani said, “he had every right to do it.”

“I need a platform to get that out,” he said Friday. “The platform requires them beating me over the head.” If that meant he had to sustain some blows, he said, so be it.

Mr. Giuliani said he was flooded with responses to his interview, calling it his “best appearance on television,” and noting that he “laid out the hypocrisy.”

Mr. Giuliani said that enduring Mr. Cuomo’s aggressive questioning was the price he was willing to pay to give voice to his claims that the Obama White House asked the Ukrainians to “dig up dirt” on Mr. Trump and Mr. Manafort.

“I didn’t talk to the president about it before,” he said. “I didn’t talk to him about it after. I won’t tell you if I talked to him about it today, or not.”