The Uruguay international posted a message on an Instagram story which contained a "racist" term - one which supposedly has affectionate undertones in Cavani's homeland - following his heroics in United's victory over Southampton at the weekend.
However, the FA are investigating the post - which Cavani has since deleted and apologised for - but Neville has since urged the authorities to properly fund diversity and inclusion training to avoid such controversies in the future.
Speaking to Sky Sports News, Neville said: "The Premier League, the FA, the EFL, the LMA and the PFA - there's five of them. They're all properly funded. They've all got lots of money. Put a diversity and inclusion set of protocols in place that are fixed.
"At least collaborate on issues that are societal. Let's drive it out of the game, root it out. It's got to be done by education and training.
"That means that players have to be part of biweekly, monthly training on diversity and inclusion: why people take offence to certain words and others don't. If it offends someone, it's a problem.
"To me, this is going to continue to happen and we're not going to be able to deal with it until this curriculum, this education is in place and then put it into schools so it's part of the school curriculum as well.
"Why is Edinson Cavani not subjected to diversity inclusion training, education the minute he comes into this country? Not to remind him of his duties but to educate him.
"If we're truly going to root out this problem of discrimination in this country it has to become part of the curriculum at school, it has to be part of a curriculum in sport.
"This could have been prevented. We're still going to see incidents like this where Edinson Cavani probably at 5pm when he reposted that Instagram story, didn't have a clue he was doing anything wrong. In 24 hours he's having to apologise for it."
Cavani came off the bench to net a brace of headers in United's 3-2 win on the South Coast, and the forward will be hoping to line up against his former side Paris Saint-Germain in the Champions League on Wednesday. body check tags ::