Monday's showdown at Hofstra University was the most-watched debate for president in history, and now that it's over Hillary Clinton and Donald Trump are jousting from afar again. Josh Robin reports.

About  80 million people tuned in to Monday night's debate and by Tuesday morning both sides claimed victory.

"Oh yes! One down, two to go!" Clinton exclaimed at a campaign event.

"It was a fascinating period of time," Trump said. "And I think we did very well."

It will be days before reliable polls show how voters feel.

In the meantime, Clinton continues an attack she made at the debate's end.

"One of the worst things he said was about a woman in a beauty contest," Clinton said.

Her name is Alicia Machado, winner of the 1995 Miss Universe, which Trump then owned. She now stars in a Clinton campaign video.

After her win, Machado gained weight the Republican nominee says now she was "a real problem."

Talking to Howard Stern at the time, he said this:

"She gained about 55 pounds, in a period of about nine months, she was like an eating machine." 

STERN: "What does a girl eat? In less than a year..."

TRUMP: "She ate a lot of everything."

STERN: "I mean, her bowel movements. How many did she have a day?"

Trump says that at the debate, he mulled raising Bill Clinton's infidelity, but decided against it because Chelsea Clinton was in the audience. A top Trump surrogate — former Mayor Rudy Giuliani — says he wouldn't have relented.

"What that woman standing there did to Monica Lewinsky — trying to paint her as an insane young woman," said Giuliani, whose own failed 2008 campaign for president didn't even last until Super Tuesday. "If you didn't know the moment Monica Lewinsky said that Bill Clinton violated her, that she was telling the truth, then you're too stupid to be president."

Amid a public rift — Giuliani ended his second marriage for a woman who is now his third wife.

Giuliani also told Politico said if he were the nominee, he would boycott the remaining two debates unless the moderator acts quote "like a journalist and not an incorrect, ignorant fact checker,"

After Monday's tension, civility is expected for a night only October 20. Both Trump and Clinton RSVP'd yes for the Catholic Archdiocese's annual Al Smith dinner in Manhattan.