He was taken ill on Tuesday while visiting the country to support his daughter Tracey, who is the England netball coach, during the World Cup.
England Netball confirmed the 65-year-old died in hospital in Sydney on Friday evening local time.
Mr Neville was a former commercial director of Bury Football Club.
The League One side said he was in Sydney with his wife, Jill, who is also club secretary for Bury FC.
In a statement, the club confirmed Mr Neville's death "with deep regret and extremely heavy hearts", and said it would pay tribute at a future date.
"Where would this Club be without Neville? Rest in Peace big Nev," the club said, adding players would wear black armbands during their match with Doncaster on Saturday.
Mr Neville was involved in the fans' group Save Our Shakers, which was formed when the side faced bankruptcy in 2002.
His sons, Gary and Phil, both started their lengthy football careers with Manchester United in the early 1990s. Gary won 85 England caps, while Phil won 59.
Manchester United tweeted: "We are sad to learn that Neville Neville, father of Gary and Phil, has passed away. Our condolences go out to his family."
An emotional Tracey Neville led her England side out on court on Friday just hours after it emerged her father had been taken ill.
Local media reported Gary and Phil Neville were in the Sydney crowd to watch England win their first match 60-19 against Scotland on the opening day of the World Cup.
An England Netball spokesperson said: "Our immediate thoughts are with Tracey and the Neville family and we please ask that the privacy of the family is respected at this time.
"Tracey has chosen to remain with the England team for the remainder of the World Cup and has the full support from everyone at England Netball."
Neville Neville, the father of former England and Manchester United footballers Gary and Phil, has died in Australia.