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The National Commission for Women issued the deadline to the popular Indian celebrity after comments he made during a press conference on Saturday.
Khan told reporters that, after several hours of filming in the wrestling ring for his upcoming film Sultan, he walked out like a "raped woman."
Commission chief Lalitha Kumaramangalam demanded an apology, telling CNN that the comment was a manifestation of India's "patriarchal attitude."
"Given the fact that he's such a public and popular figure, especially with someone of his standing in this country, especially when he's seen as an icon, it's unfortunate [and] reprehensible," she said.
"We're not trying to conduct a witch hunt. I only want him to apologize. Nobody is asking him to grovel."
Kumaramangalam told the Press Trust of India that Khan would be issued with a summons to explain himself before the commission if he failed to respond.
'To err is human, to forgive divine'
"If I were to pick up a 120-kilo person and drop him down I would do it 10 times -- in a wrestling match it happens once, twice," he said.
"Ten times from five different angles, so for six-and-a-half to seven hours either I'm picking up and I'm throwing or he's picking up and he's throwing. So it is like the most difficult thing. When I walk out of that ring it is actually like a raped woman walking out."
As soon as he made the comment, Khan appeared to regret his choice of words. "I don't think I should have [said that]," he said immediately afterwards. "For me, it's like the most difficult. I couldn't take steps."
Taking to Twitter to defend his son, scriptwriter Salim Khan said while what Salman said was wrong, he hadn't intended to offend.
"Nevertheless I apologize on behalf of his family his fans & his friends. Forgiveness is to pardon the unpardonable or it is no virtue at all," he wrote. "To err is human to forgive divine."