"There is no question over any of the players," Vidic told the BBC. "They did everything that was asked of them.
"Sometimes you do things well, sometimes you don't. But the desire was there. They tried really hard.
"From the first day until the last you didn't hear any player say anything bad about the training. That shows the players have respect for the manager."
Moyes was sacked in April, only 10 months into a six-year contract, after United failed to qualify for the Champions League for the first time since 1995/96.
Vidic, who is set to join Inter Milan after playing the last of his 300 games for United against Southampton on Sunday, said there were no dressing-room fallouts with Moyes and the only arguments were between players desperate to improve.
"I am not saying we have had a good season. It was a bad time and the players could have done better. But everyone lost belief in the team," the 32-year-old Serb said.
"The players did argue among ourselves. But we were arguing to get better. When you have bad times, people show they care. It hurts but we could say those things to each other because we have been together for so long.
"We had some hard moments in the dressing room but we never argued with the manager or his assistants. Never."
United are set to appoint a new permanent manager, with Dutchman Louis van Gaal the favourite to land the role.