Ferguson said Beckham became too obsessed with fame in the latter years of his Old Trafford career and reiterated his claim that Rooney asked to leave the club at the end of last season.
He confirmed that Keane, his long-serving captain, was asked to leave the club after lashing out at several of his younger team-mates in an interview with in-house television channel MUTV that was never broadcast.
He also revealed that he was twice offered the England manager's job, in 1999 and 2001, but turned it down.
"There was no way I could contemplate that. It wasn't a bed of nails I was ever tempted to lie on," he said in 'My Autobiography'.
Ferguson, 71, stepped down as United manager in May after a record-breaking 26-and-a-half-year tenure in which he won 13 Premier League titles, two Champions League crowns, five FA Cups and four League Cups.
The Scot developed a reputation for imposing unyielding discipline on the players in his squad and he was particularly forthright about several of the superstars he worked with.
He described Beckham as "the only player I managed who chose to be famous, who made it his mission to be known outside the game".
Ferguson said he decided to sell Beckham to Real Madrid after an incident following an FA Cup loss at home to Arsenal in February 2003 when he accidentally kicked a boot into the midfielder's face during a post-match argument.
"The next day the story was in the press. In public an Alice band (worn by Beckham) highlighted the damage inflicted by the boot," Ferguson said.
"It was in those days that I told the board David had to go. The minute a Manchester United player thought he was bigger than the manager, he had to go. That was the death knell for him."
Expanding on the story at a press conference in central London to mark the launch of the book, Ferguson said that Beckham's focus shifted after he married the pop star Victoria Adams in 1999.
"The big problem for me is that he fell in love with Victoria. That changed everything," he said.
Ferguson said that his relationship with Keane broke down irretrievably after the Irishman "slaughtered" players including Kieran Richardson, John O'Shea, Darren Fletcher and Rio Ferdinand in an MUTV interview in 2005.
Ferguson recalled an exchange with his assistant manager, Carlos Queiroz: "'He needs to go, Carlos,' I said. 'One hundred percent,' he said. 'Get rid of him,' I said."
Ferguson described Ronaldo, who left United for Madrid in 2009, as "the most gifted player I managed" and said watching him play for the first time was "the biggest surge of excitement... I experiencied in football management".
The Scot disclosed the contents of the conversations he had with Rooney on the two occasions he asked to leave the club, in 2010 and in May this year.
On the first occasion, Ferguson said Rooney was worried about United's ambition and urged him to sign Germany playmaker Mesut Ozil, who went on to join Arsenal in a club-record transfer last month.
"Wayne said that we should have pursued Mesut Ozil, who had joined Real Madrid from Werder Bremen," Ferguson said.
"My reply was that it was none of his business who we should have gone for. I told him it was his job to play and perform."
Ferguson said Rooney "seemed to tire in games" during his last campaign and claims the former Everton striker asked to be allowed to leave.
"He came into my office the day after we won the league and asked away," Ferguson said.
"He wasn't happy with being left out for some games and subbed in others."
At the press conference, Ferguson reiterated his support for his successor, David Moyes, who has overseen United's worst start to a league season since 1989.
"You know something Manchester United are the only club in that league that can win the league by coming from behind, believe me," he said.
"Once they get the ball rolling, they'll be fine. David Moyes is in a great position. There's a great support system."
The book, which follows Ferguson's previous 1999 autobiography, 'Managing My Life', goes on general sale on Thursday.
Quotes from Alex Ferguson's autobiography
Some quotes from former Manchester United manager Alex Ferguson's autobiography, which will be released later this week:
On former Manchester United midfielder David Beckham:
David was the only player I managed who chose to be famous, who made it his mission to be known outside the game.''
On former United captain Roy Keane:
He has the most savage tongue you can imagine. He can debilitate the most confident person in the world in seconds with that tongue.''
On Chelsea striker Fernando Torres:
Torres was blessed with great cunning: a shrewdness that was borderline Machiavellian. He had a touch of evil, though not in the physical sense.''
On Liverpool captain Steven Gerrard:
I am one of the few who felt Gerrard was not a top player.''
On former Manchester United goalkeeper Mark Bosnich:
A terrible professional ... we played down at Wimbledon in February (2000), and Bosnich was tucking into everything: sandwiches, soups, steaks. He was going through the menu, eating like a horse.''
On Chelsea midfielder Frank Lampard:
I didn't think of him as an elite international footballer.''
On Napoli coach Rafa Benitez, previously of Liverpool and Chelsea:
The mistake he made was to turn our rivalry personal ... I had success on my side.''
On managing Manchester United:
I would sit in my office in the afternoon, with my work complete, wanting company.''
On player power:
In modern football, celebrity status overrides the manager's power. In my day you wouldn't whisper a word about your manager. You would fear certain death. In my later years, I would hear constantly about players using their power against managers, and the player receiving the support of the public and even the club.''
Recalling what a un-named senior Inter Milan official told him:
Do you know the difference between the English and Italians In England they don't think a game can ever be corrupt. In Italy they don't think a game cannot be corrupt.''
On the United States:
The States always intrigued and inspired me. I fed off America's energy and vastness, its variety.''
On former US President John F Kennedy:
I developed a forensic interest in how he was killed, by whom and why.''