Neville revealed Woodward "briefed" reporters before victory over Leicester City.
He told Sky Sports: "From a board perspective, it has been messy over the last couple of weeks.
"I think of the briefing Ed Woodward gave journalists ahead of the Leicester game, where he explained why United weren't going to buy centre-backs and why Jose was not being backed, trying to justify to the media why he's not backed his manager. It's too late for that, Mourinho has already been given a contract extension - back him until the end.
"My view would be he has to stay to the very end of that contract and be backed to the end of the contract. The briefing undermined Jose Mourinho, there's no doubt about that. Jose knows that and he's not going to take any messing around from anybody.
"The CEO sat in his spot for two minutes after the game on Sunday in what I can only describe as something I don't want to see the CEO of Manchester United do. At the end of the game, stand up, go and shake the opposition CEO's hand, tell him well done and that you'll see him later in the season, smile and be bigger than anything.
"All I could see was a show of frustration, or something that could also be interpreted as another show of dissent."