"I regret any hurt caused as the same was inadvertent and unintended," Sinha said a day after he compared rape and betting in sports.
"I reiterate my deep sense of regard and respect for women, and commitment to gender issues," the Central Bureau of Investigation (CBI) told the media.
Speaking in support of legalising betting here Tuesday, Sinha had said: "What is wrong if we legalise betting? Do we have the enforcement agencies (to stop betting)?
"If you cannot enforce the ban on betting, it is like if you can't prevent rape, you enjoy it."
The remark caused nationwide furor. On Wednesday, National Commission for Women (NCW) chief Mamata Sharma demanded his resignation.
"I am sorry to say (that) an agency, which is there for women's security, the head of that institution is saying this. If he has said so, he should resign.
"We thought we had to sensitise people at lower levels, but this example shows that people at higher level also have to be sensitised," she said.
Political parties too were livid.
Bharatiya Janata Party spokesperson Nirmala Sitharaman called the comment "shocking".
"Wonder if his colleagues in the bureau, his family and well-wishers approve of his view," she asked.
Added Brinda Karat of the Communist Party of India-Marxist: "How can he even think of using such terms, this is absolutely outrageous."