Woodward, the architect of England's 2003 World Cup triumph in Australia, had been installed as clear favourite to become the first incumbent of the new role.
But Woodward, 55, ruled himself out of the process this week and said he was committed to his role at the British Olympic Association.
That has angered former England captain Cotton, who believes Steele must carry the can.
"This is where the (RFU) chairman and non-executive directors have got to step in," he told BBC Radio 5 Live's Sportsweek programme.
"The only way it will be solved is by John Steele's removal, he has shown he is not up to the job.
"If it's a choice between John Steele as chief executive or Clive Woodward as performance director, it's a no-brainer.
"I think he should go - look at the publicity the RFU have had over this, he's brought them into disrepute, you've got your major candidate walking away, it can't get much worse.
"It's probably the most important role in English rugby and world rugby and he's talking about downgrading the role to bring in Mr Average instead of Clive Woodward or Graham Henry, the people England need."
The performance director position has proved a hot topic of debate in recent weeks at Twickenham and beyond, and when the initial job description was unexpectedly diluted in terms of its powers, many doubted Woodward would retain interest.
The RFU management board, on a narrow majority vote, then effectively reinstated the old job specification, with current England manager Martin Johnson reporting directly to the performance director after the World Cup in New Zealand later this year.
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