The coronavirus outbreak has led to the suspension of every major league in Europe, including the Premier League, as well as UEFA club competitions.
United sit in fifth place in the league table with nine games to go and remain in contention to win the Europa League and FA Cup but Shaw said the campaign should be made "void."
"Scrap it and start again," he said during a Combat Corona Twitch FIFA competition. "Start it again, yeah.
"It's got to be, you know. If we can't carry it on, it's got to be void."
Former United defender Rio Ferdinand also shared the view that the season should be cancelled, and that would mean rivals Liverpool, who sit 25 points clear at the top of the table with 10 games remaining, would miss out on a first title since 1990.
UEFA president Aleksander Ceferin has said that leagues that end early without permission could face a European competitions ban next season.
Belgium became the first European top-tier league to cancel the rest of its season last week with Club Brugge crowned champions.
There have been suggestions that the rest of the campaign could be played behind closed doors but Shaw said he would be against playing without supporters.
"Fans are so important," he added. "You realise it even more [now]. I think the sport is for fans really. I think if you don't have fans, and you don't play in front of fans, it just doesn't feel right.
"Especially on matchday, the fans are always amazing and always help the team. Whether it's home or even away, our fans are always brilliant and I feel like they're always there with us."