Gold offered his support to the Manchester United executive vice-chairman after his home was targeted by angry protesters last month.
Police are investigating the attack after videos emerged on social media showing a group hurling flares towards Woodward's mansion.
"I know exactly what Ed's experienced. I've experienced it four or five times myself," Gold said.
Gold and co-owner David Sullivan were forced to leave the directors box amid chaotic scenes during a Premier League defeat by Burnley at the London Stadium in 2018.
"The league will come together as a unit to protect the likes of Ed and punish those perpetrators. It cannot be tolerated.
"United are a big club, Ed Woodward is a big figure in the game. He can't be worried about his wife and children."
West Ham, who visit champions Manchester City on Sunday, are locked in a relegation battle this season and sit 18th, with fans blaming the owners for their recent struggles.
"The story is we're liars, we've taken all the money. That's wrong ... I've not received a penny in salary or expenses from West Ham in 10 years," Gold said.
"All my young life, the club was in the old second division. We're in the Premier League now. OK, we've got a few problems but overall we're moving upwards."
Additional reporting by Reuters