The former captain is enraged that the Red Devils have yet to confirm or deny reports that the manager is to be removed from his position
The manager is set to be sacked by the Old Trafford hierarchy following a disastrous debut season, with Sunday's 2-0 loss to Everton confirming the club will miss out on the Champions League next term.
Neville, who spent his entire playing career at United, questioned the lack of an official comment from the club in response.
"I think the whole situation at the club right now is repulsive," Neville told Sky Sports. "The club really need to make a statement.
"It's been a world of madness where the average manager gets sacked every 12 months and I've always felt Manchester United should be different and hold itself up as club that stands against what is happening in the game.
"We used to laugh at what happened in Italy 20 years ago and say it is ridiculous but now we've almost become accustomed to it.
"I think fans are well within their rights [to call for Moyes to go], but I don't like it when professional people from inside the game come out and suggest the manager should be sacked.
"It isn't something you'll hear from me. I've never done the job. I do fear for David Moyes because of the explosion that has happened today, but I'm hoping the club will come out and clarify it and it won't be true."