The 54-year-old took over at Old Trafford in 2013 after Ferguson stepped down, but lasted only 10 months in the job before he was sacked.
In September, the Portuguese head coach claimed that United "failed to evolve" under Moyes and Van Gaal, a claim that the Scot has rejected.
"Jose has his opinion, but there was always going to be a transitional period after Sir Alex's time was over," Moyes told The Mirror.
"There was a changing of the guard. Not only did I replace Alex, David Gill had also stepped down as chief executive and Ed Woodward came in. So it was always going to take time. Yes, I took over the champions, but clubs like Chelsea and Man City spent huge amounts of money that summer in a bid to overtake us.
"Having said that, I accept totally that when you are manager of Manchester United, you have to win. I didn't win enough. But I don't think there is a manager out there who would have been able to do something better or quicker in the time I was given after Sir Alex retired. It was all about small margins."
United finished seventh in the Premier League table when Moyes was relieved of his duties with four games remaining in the 2013-14 campaign.