Scholes has also revealed he does not expect to be involved in a coaching capacity at Old Trafford next season.
Woodward has endured a rocky first year since assuming his current position, with the ill-fated appointment of David Moyes as manager, some frustrating summer transfer experiences and the team's disappointing on-field results.
But Scholes feels Moyes has been made too much of a scapegoat for the poor 2013-14 season and more questions need to be asked of Woodward.
Scholes said: "David Moyes took a lot of stick, but I believe he's a top manager. I'd question if 10 months was enough time.
"Edward Woodward has an awful lot to prove this time that he's good enough at his job. He has to bring the players in that the new manager wants.
"It's obvious that last year he didn't manage to do that. If he doesn't we are not going to get anywhere near the top."
Scholes, 39, retired a year ago but was brought back to United in a coaching role for the final four games of last season by interim manager Ryan Giggs.
He has not been told whether he will continue under new manager Louis van Gaal next season, but is not expecting to be retained.
Scholes said: "I've not spoken to Edward Woodward. I came back for Ryan Giggs for the last few games of last season to try to help out, but I'm not waiting for a phone call and don't expect to be at United next season."
Scholes went on to say how he believes England can get the best out of his former United team-mate Wayne Rooney at the World Cup this summer.
Scholes, who played in two World Cups, has urged manager Roy Hodgseon to make sure Rooney channels all his energy into his responsibilities up front.
Scholes said: "Wayne wants all the responsibility to score. He'll try to play left-back, right-back. Sometimes he does that too much instead of saving himself and his energy for what his teams need - the ball in the net.
"He needs to use his energy more effectively now as he's a player who likes to be up front on his own and I don't think he's great with partnerships.
"To get the very best from Wayne in Rio, the manager needs to tell him, 'Don't bother running back. Stay up top. Stay centre forward. Score goals. That's your job in my team'."
Scholes also voiced fears that 28-year-old Rooney, who burst on to the scene as a precocious teenager with Everton, may already have passed his peak.
He said: "Wayne was in the Everton team at 16 years of age, in 2003.
"Since then he's played at Euro 2004, two World Cups, Premier League, and Champions League every year at United.
"There's a chance he's worn out. Wayne's peak may have been a lot younger than what we'd expect of footballers traditionally.
"Age 28 or 29 has been the normal 'peak'. With Wayne, it could have been when he scored 27 league goals in 2011/2012 when he was 26."