Twickenham, situated in the southwest district of greater London from which it takes its name, has been the home of English rugby for over a hundred years.
But from September, the 82,000-seater ground will be known officially as the Allianz Stadium in a deal worth more than £100 million ($127 million, 116 million euros).
RFU chief executive Bill Sweeney has denied the English governing body has "sold out" but Woodward, the coach of England's 2003 World Cup-winning squad, lamented the decision.
"In a wonderful summer of sport, the decision to rename Twickenham to Allianz Stadium is an almighty blow," he wrote on his LinkedIn account on Wednesday. "The reason is obvious: money.
"The RFU badly needs cash. English rugby and the RFU has fallen off pace in world rugby and compared to other domestic and global sports."
Woodward, who also played international rugby at Twickenham as an England centre, added: "The RFU board sold its soul by renaming this historic stadium. Many, including myself, question why and how it has come to this?
"Some say a name means nothing but Twickenham represents 100+ years of heritage.
"Those questioning this are called 'traditionalists' by the RFU which is a cheap shot and no different than calling people 'old farts' as (former England captain) Will Carling did years ago."
The failure to even include 'Twickenham' in the new title, along the lines of other sponsorship agreements such as the Emirates Old Trafford cricket ground in Manchester, has angered many rugby fans.
But with the RFU having forecast a loss of over £40 million for the 2023/24 financial year, the English game could benefit from a cash injection at a time when it is still reeling from the collapse of three Premiership clubs in Wasps, Worcester and London Irish.
Woodward, however, warned renaming Twickenham was "no silver bullet".
"The RFU must restore England to the pinnacle of world rugby," said the 68-year-old.
"Money is needed, but this naming rights deal is no silver bullet, the total removal of Twickenham from the venue represented a very sad and poignant day.
"It's impressive how much money Wimbledon leaves on the table each year. They haven't sold the naming rights to The Championships or Centre Court, they understand the brand value Wimbledon represents.
"Can you imagine Novak Djokovic and Carlos Alcaraz playing on Allianz Centre Court? How would Scottie Scheffler feel arriving at Allianz Augusta? Do we want to see the first ball in the Ashes at Allianz Lord's? Or see England at Allianz Wembley?"
