The ex-premier also discloses that he found the incident when the then deputy prime minister thumped a voter in the 2001 election campaign "extraordinarily funny".
Mr Blair also heaps praise on John, now Lord, Prescott saying he was lucky to have him as a deputy.
In his book, A Journey, Mr Blair recalls bumping into Charles after his first meeting with the deputy PM.
"Does he ever do that thing with you?" asked the Prince.
"When he's sitting opposite you, he slides down the seat with his legs apart, his crotch pointing a little menacingly, and balances his teacup and saucer on his tummy.
"It's very odd. I've never seen someone do that before. What do you think it means?"
Mr Blair says he replied: "I don't think it means anything, really."
Charles continued: "Hmm. You don't think it's a sort of gesture or sign of hostility or class enmity or something?"
Mr Blair reassures him that his deputy made the same gesture to him quite often, adding finally: "I think he just likes drinking his tea that way."
Of the punch thrown at a protester who had just hit the then Mr Prescott with an egg, Mr Blair writes: "I personally felt the thing was extraordinarily funny."
But he concedes there was another point of view within the party "and some of the women in the operation were voicing it loudly" that a deputy PM should not behave in such an undignified way.
Mr Blair says he rang Mr Prescott to ask him to apologise, only for his deputy to reply: "I'm bloody well not. So you can forget it."
Mr Blair also reveals Mr Prescott unleashed a four-letter tirade after reports a senior Liberal Democrat might enter the Cabinet.
But he concludes his deputy was "a one-off. Occasionally my bane. More often my support. But genuine, unvarnished and, in the ultimate analysis, true. And in my profession, you can't say better than that".
Mr Blair said he should have sacked Mr Prescott after the April 2006 revelation of his affair with his diary secretary, Tracey Temple.
But he could not bring himself to do it, partly out of a feeling that he should "do right" by the deputy prime minister's wife, Pauline.
He wrote: "In purely selfish terms it would have been better to fire him, I knew that.
"It would have given the media their scalp. It would have allowed some change at the top and, even if that had turned into a TB/GB (Tony Blair/Gordon Brown) contest, it would have served to flush people out. But I just couldn't bring myself to do it."
Mr Blair speculated that politicians took the risk of having an extramarital affair as a release from the "supreme self-control" needed to get to the top.
He said: "There is the moment of encounter, so exciting, so naughty, so lacking in self-control.
"Suddenly you are transported out of your world of intrigue and issues and endless machinations and the serious piled on the serious, and just put on a remote desert island of pleasure, out of it all, released, carefree."
He said politicians became "extraordinarily, incomprehensibly naive" in these circumstances and were open to romantic approaches from "the first person who appears to take an interest... who, above all, doesn't think, act or talk like a fellow politician".
Asked about his encounter with the Prince of Wales, a laughing Lord Prescott told BBC Radio 4's World At One: "There was nowhere to put the cup and saucer, so I rested it on my stomach.
"I believe he told a friend 'I have never seen anybody doing limbo tea-drinking before'."
And Lord Prescott thanked members of Mr Blair's constituency party for helping the PM make up his mind to back his deputy over punching the protester.
In his book, Mr Blair wrote: "When the news had come on in the Trimdon Labour Club, everyone had cheered as JP hit the guy. That settled it."
Lord Prescott said: "Thank you Trimdon. You helped Tony say 'John is John'."
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