Moyes succeeded Sir Alex Ferguson as manager last summer on a six-year contract but was sacked last month, United eventually finishing 22 points behind table-topping Manchester City and failing to qualify for Europe.
The last of Vidic's eight-and-a-half years with the club, ahead of a move this summer to Serie A side Inter Milan, was a disappointing one.
"People end up saying the players are not good enough and we need to buy better ones," he told BBC Sport
"It was a bad time and the players could have done better
But everyone lost belief in the team."
Moyes' tactics were often criticised, notably after the 2-2 draw with relegated Fulham when United racked up 81 largely aimless crosses before conceding a stoppage-time equaliser.
Vidic told the Daily Telegraph: "I am not saying that the David Moyes way was bad, but these players feel more comfortable playing a certain way.
"(He) lost his job because he did not succeed in doing what he wanted to achieve
He was really committed to the job and desperately wanted to do well
But unfortunately, it didn't happen and we are all sad."
The Serbia international acknowledges strong words were exchanged but denies any were aimed at Moyes
"The players did argue among ourselves," he told BBC Sport
"But we were arguing to get better
"We never argued with the manager or his assistants
Never
"People said we lost faith in David Moyes
We didn't
We knew he was trying to build something."
Meanwhile, former United captain Roy Keane has read the riot act on Vidic's fellow defenders Phil Jones and Chris Smalling, insisting the pair have "gone backwards"
Jones and Smalling were both named in England's squad for the World Cup but Keane believes they have failed to live up to expectations and time is running out
"We were told two or three years ago Jones was going to be the new Duncan Edwards and Smalling was this and that," Keane said at the launch of ITV's World Cup coverage.
"I've watched United live nine or 10 times this year and they have been none of those things - if anything I think they have gone backwards.
"Jones needs to toughen up - he's playing for Man United
Every time I see him he is getting carried off
"I don't think they'll play (for England) on their current form
The two of them have had disappointing seasons."
Keane believes Moyes' sacking was harsh and says more blame should lie with executive vice-chairman Ed Woodward, who failed to secure several high-profile targets last summer.
"Why give Moyes a six-year contract? He had one transfer window and it's not always down to the manager when players don't come in," Keane said.
"I think Ed Woodward needs to look at himself
He's got to get deals done
I think Moyes should have been given more time."
Source : PA
Source: PA