Moyes lasted less than a year as manager of the Premier League giants before he was sacked in April after a series of poor performances.
But Keane said the players needed to do more for Moyes, particularly as the Scot tried to replace the retired and successful Sir Alex Ferguson at Old Trafford.
"I wonder about the current United dressing room," Keane wrote in his autobiography, The Second Half.
"When a manager like Sir Alex Ferguson is replaced, the new man needs a helping hand along the way.
"Does that mean every player should like the new manager or his coaching staff, or love his new sessions and everything about him? No.
"I look at the current players, and they should have been doing a lot better.
"It might be argued that it was up to the manager to motivate them. But not liking a manager, for whatever reason, can never be an excuse for not going out and doing your best.
"Looking at what happened to David Moyes, I have to conclude that he can't have had a strong dressing room: he had a weak dressing room."
On Ferguson, Keane - who left United in November 2005 – was furious with the way his departure from the club played out.
Keane slammed Ferguson and former United chief executive David Gill.
"I'd known for a few days they were trying to get rid of me," he said.
"I said to Ferguson 'Can I play for somebody else?'
"And he said 'Yeah, you can cos' we're tearing up your contract.'
"I knew there'd be clubs in for me when the news got out. I said 'Yeah, I think we have come to the end.'
"I just thought 'F****** p****s' - and I stood up and went, 'Yeah, I'm off.'"
Keane also said he had no regrets over his reckless tackle on Alf-Inge Haaland's and revealed he once headbutted Peter Schmeichel.
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