Three actors donned the Palestinian keffiyeh scarf during the curtain call on the opening night of the company's production of Chekhov's The Seagull on Saturday.
The move sparked criticism from the Jewish community and triggered the resignation of prominent board member Judi Hausmann after 16 years in the role.
Ms Hausmann, who has been contacted for comment, chaired the company's Remix organising committee and was one of its foundation directors.
The STC said it was deeply sorry for its initial response to the episode, in a statement issued late on Wednesday.
"We understand the actions at the curtain call and our immediate response has hurt many in our community," the statement said.
"We believe that it was not the intent of the actors involved to cause any harm or offence."
Other cast and crew members did not know the actors planned to wear the keffiyeh beforehand, the STC said, and it has told performers they are free to express their opinions and views on their own platforms.
The company said it recognised that while actors bring their whole selves onstage, audiences come to the theatre to "experience the content in that play and that play only".
"We support individual freedom of expression but believe that the right to free speech does not supersede our responsibility to create safe workplaces and theatres."
Earlier on Wednesday, the National Council of Jewish Women Australia said the STC was loved and supported by many in the community.
"You have failed on a deep and moral level. We cannot forget this easily," the council said in an online post.
Daniel Grynberg, a long-time STC subscriber and former head of fundraising organisation JCA, weighed in on the protest.
Mr Grynberg said he would have felt sickened had he been in the audience, because of threats to his family in Gaza and recent anti-Semitism in Sydney.
"We (Jews) in Australia have felt totally abandoned by the progressive left. Our pain is to be understood or contextualised. And Israeli suffering is ignored," he wrote in the Australian Financial Review.
"So the STC actors' 'protest' is viewed by me and by most Sydney Jews in that light.
"The STC – which has done so very much to show care for so many other minorities and historically disadvantaged groups (First Nations, LGBTQIA, women's voices) – has done precisely nothing to make this particular minority feel supported."
Singer Tania de Jong wrote to the STC and described the protest as an "appalling and arrogant" stunt that threatened the entire arts sector.
The Media Entertainment and Arts Alliance said it would support any members disciplined by their employer for expressing their views.
"MEAA respects the rights of members to publicly express their views on this issue and will support any members who are subject to disciplinary action for simply engaging in freedom of expression," acting chief executive Adam Portelli said.
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