Graham spoke at some length with Nixon, dissecting the network TV coverage and briefing the president on CBS' coverage, which he deemed the most negative.
"I felt like slashing their throats. But anyway, God be with you," said Graham, who also told Nixon the speech was "his finest hour."
"CBS was knocking it but ABC and NBC were not bad?" Nixon replied.
"They were all favorable," Graham replied.
The calls are significant because they show the pressure Nixon was under and how desperate he was for validation as the crisis wore on, said Ken Hughes, a research specialist for the University of Virginia's Miller Center.
"It was one of the worst nights of his life and even two people as famously upbeat as Ronald Reagan and George H.W. Bush were unable to cheer him up," said Hughes, who studies and reviews Nixon tapes.
"He saw the writing on the wall," he said.
The tapes also include discussions between Nixon and his aides about the Vietnam War and contain a lengthy recording of Nixon and Soviet leader Leonid Brezhnev chatting warmly in the Oval Office before a historic summit in June 1973.
In a June 7 recording dealing with Vietnam, Nixon told his chief of staff, Alexander Haig, that South Vietnam's president, Nguyen Van Thieu, had leverage in the peace settlement because he knew the U.S. would be embarrassed if South Vietnam fell as soon as American forces left.
"You know, we're in a real tough position, aren't we?" Nixon said. "Thieu knows that we don't want them to go down the tubes so soon after the darn war is over, you know, for our failure, so he thinks that he's got us by the short hairs."
He added later, "It's a strange, strange world we're living in, isn't it, Al?"
The tapes also captured a rare and unscripted conversation between Nixon and Brezhnev. The two met for an hour on June 18 and chatted about personal topics, including their families and future travel plans, with only an interpreter present.
The conversation happened before the start of a historic seven-day summit that was part of Nixon's strategy of detente with the Soviet Union.
"We must recognize, the two of us, that ... we head the two most powerful nations and, while we will naturally in negotiations have some differences, it is essential that those two nations, where possible, work together," Nixon told Brezhnev.
"If we decide to work together, we can change the world. That's what — that's my attitude as we enter these talks."
The conversation is remarkable because of the camaraderie that is evident, said Luke Nichter of Texas A&M University-Central Texas in Killeen, who runs a website cataloging Nixon's secret recordings. Both men discuss their children and Brezhnev even talks about his grandson's attempts to pass college entrance exams.
"These are Cold War archenemies who are talking like old friends," he said. "This is very unusual."
The recordings were released at the Nixon Presidential Library and Museum in Yorba Linda and cover April 9, 1973, to July 12, 1973, the day before the existence of the covert recording system was revealed to a Senate committee probing Watergate.