The Gunners on Sunday surprisingly went public with their backing of manager Mikel Arteta, who had strongly criticised the officiating and decision to allow Anthony Gordon's match-winning goal in a 1-0 defeat by Newcastle at St James' Park on Saturday evening.
The effort was not ruled out despite three separate lengthy VAR checks, with Joelinton's challenge on Gabriel before Gordon turned home from close range a particular point of contention for Arsenal amid complaints that the ball also may have gone out in the build-up.
A furious Arteta said he "felt sick" and was "ashamed" at the "disgraceful" and "embarrassing" decision to award the controversial goal, with Arsenal then releasing a statement saying they "wholeheartedly supported" the manager's comments.
The Gunners added: "The Premier League is the best league in the world with the best players, coaches and supporters, all of whom deserve better.
"PGMOL urgently needs to address the standard of officiating and focus on action which moves us all on from retrospective analysis, attempted explanations and apologies.
"We support the ongoing efforts of Chief Refereeing Officer, Howard Webb and would welcome working together to achieve the world-class officiating standards our league demands."
Addressing Arsenal's statement during his appearance as a co-commentator on Sky Sports' Monday Night Football programme ahead of the London derby between Tottenham and Chelsea, Neville said: "I thought the statement was really poor.
"I thought Liverpool had a shocking one go against them a few weeks ago here and we knew it was wrong at the time. They had a lot to feel hard done by, but I thought the day after when Liverpool wrote their statement mentioning sporting integrity and that they were looking at all options and then the suggestion of a replay. I thought that was, to be fair, quite dangerous.
"I think the Arsenal statement is quite dangerous. I think all clubs have signed up to a new behavioural charter at the start of the season. I’m not saying that referees shouldn’t be put under pressure that Mikel Arteta or other managers shouldn’t feel aggrieved at the end of matches.
"What I am saying is that the clubs themselves should definitely behave better in these circumstances. I am looking at the Premier League now really and the leadership in the Premier League because the reality of it is, they have got to start to protect referees.
"At the start of the season, I was excited by the new transparency, the apologies to managers and the fact that the referees were communicating with the clubs. But in return, what they are getting is significantly and seriously undermined by their own clubs, and I think the Premier League and the other clubs to be fair have to get the clubs in order who are writing these letters.
"I think the Arsenal statement is quite dangerous... they should behave better than that"
"I think this is uncharted territory, the idea of these letters being written. To me it feels like Mikel Arteta has gone off on one on Saturday, which is fine, he’s emotional, and he’s almost gone to his board on the Sunday and said you need to back me and they have fallen for it.
"Arsenal as an institution, as a football club, are a massive, great, wonderful football club with huge history. They should behave better than that. They will feel disappointed I think in a few years, whoever it is that has sanctioned that statement.
"I do think that the Premier League have to come down on their clubs. I know the Premier League is the clubs, but I do think they have to now really say to the clubs - 'come on, we expect a lot better than this.'"