The RFU had been looking for additional investment and, in a document seen by the BBC, was considering selling their base in Twickenham and buying a 50% share in Wembley.
Those plans were shelved in favour of redeveloping the stadium, which had been quoted at a cost of £663m by the Twickenham Stadium Masterplan Programme (TSMP).
Woodward said Twickenham should be considered in the same breath as other iconic sporting venues such as Lord’s, Wimbledon and Wembley whose names are untouched by commercialism.
“They haven’t sold the naming rights to The Championships or Centre Court, they understand the brand value,” said the 68-year-old, who coached England between 1997 and 2004.
“I take no comfort that Ireland, Wales, and Scotland sold their stadium naming rights. England should not feel proud.”
Asked how much the deal with Allianz was worth, Sweeney said: “It’s a multi-year commitment and it’s well over £100m, so it’s a fair value for what you’d expect for stadium naming rights these days.”
London Irish, Wasps and Worcester all went into administration in the 2022-23 season.
Sweeney added: “It’s money that we need to invest back into the game.
“We’ve had a tough time coming out of Covid, the game’s under pressure – we saw three professional clubs go out of business [and] that was extremely painful.
“So, this is money we need to generate, to invest back into the community game so we keep that pipeline going.”
The inclusion of Twickenham increases Allianz’s portfolio to eight stadiums around the world.
The venue will host the final of the 2025 Women’s World Cup, while the first international fixture under the new name will see England’s women play world champions New Zealand on 14 September.