Ian Murray said he would step down from his role so the organisation can “rebuild its reputation”.
The society had issued a strongly-worded statement following Harry and Meghan’s interview with Oprah Winfrey in which it said it was “not acceptable” for the couple to make claims of racism in the press “without supporting evidence”.
And it went on to say the UK press was not racist.
But since then, the body has been heavily criticised.
Mr Murray said: “While I do not agree the society’s statement was in any way intended to defend racism, I accept it could have been much clearer in its condemnation of bigotry and has clearly caused upset.”
He insisted the press had a “proud record of calling out racism”.
However, a new statement from the society’s board admitted there was “a lot of work to be done in the media to improve diversity and inclusion.
“We will reflect on the reaction our statement prompted and work towards being part of the solution,” it added.
Mr Murray said as executive director, he should “take the blame” and announced he was to step aside.
He added the original statement was “not intended to gloss over the fact the media industry in the UK does have work to do on inclusivity and diversity”.
More than 160 journalists of colour and the editors of the Guardian, Financial Times and HuffPost UK previously issued statements saying they did not agree with the Society of Editor’s position.
Guardian News and Media editor-in-chief Katharine Viner said on Tuesday: “Every institution in the United Kingdom is currently examining its own position on vital issues of race and the treatment of people of colour.
“As I have said before, the media must do the same.
“It must be much more representative and more self-aware.”
HuffPost UK editor-in-chief Jess Brammar also tweeted to say she disagreed with the statement.
She wrote on Monday: “I considered not saying anything about this because I’m aware I won’t make myself popular with my peers, but I’m just going to stand up and say it: I don’t agree with statement from my industry body that it is ‘untrue that sections of UK press were bigoted’.”
ITV News presenter Charlene White also said she would not host this year’s Society of Editors’ Press Awards because of the row over the comments on Meghan and Harry’s interview.
Alison Gow, president of the Society of Editors, thanked Mr Murray for “his tireless work on behalf of the society”.
She added it was “committed to representing all journalists and upholding Journalism; I am clear on what our mission must be, and we will strive as an organisation to listen and hear everyone’s views, and be strong advocates and allies for all those we represent.”