Her comment came a week before decisions from Northern Ireland prosecutors on whether 17 soldiers involved in the 1972 Bloody Sunday shootings in Londonderry will face prosecution.
Ms Bradley has since pledged to rebuild the trust of victims hurt by her comments, making it clear she would not resign and that she wants to deliver for the people she had offended.
"I am determined to prove myself by delivering for the people of Northern Ireland," she said.
"It is an enormous honour to be Secretary of State for Northern Ireland, it's an enormous honour to serve the people of Northern Ireland and Government, and it is something I really want to deliver on."
Ms Bradley added: "I shouldn't have said it and I want to say sorry to all those people, all those families that have been kind enough to share their experiences with me.
"I want to say sorry to them because I didn't want to cause hurt or pain or distress to them in any way, and what I want to do is deliver for them, and I am absolutely determined I will do.
"I recognise that a slip of the tongue at the wrong moment has caused enormous distress.
"I want to be very clear - I do not believe what I said, that is not my view.
"I believe that where crimes have happened, no matter who the perpetrator, they should be properly investigated by an independent authority and they should be prosecuted.
"There is no excuse for anybody where a crime has been committed."
However, a number of families bereaved by state violence did accept an invitation to meet Mrs Bradley at Stormont House on Friday morning.
Ahead of the private meeting, campaign group Relatives for Justice tweeted a picture of the delegation, saying: "A delegation of families bereaved by the British army and RUC outside Stormont House on way to meet Karen Bradley, look her in the eye, and tell her what they think."
A delegation of families bereaved by the British army and RUC outside Stormont House on way to meet Karen Bradley, look her in the eye, and tell her what they think .@FinucaneCentrepic.twitter.com/MBdBf0d1vl
Speaking after the meeting, Frances Meehan, whose brother was shot dead by the British Army in 1980, called for Mrs Bradley to resign:
"I wanted to meet her because I wanted to look her in the eye to tell her how I felt about her comments in the House of Commons.
"I also wanted to say to her that on this day, International Women's Day, that she is an insult to women. We know she has apologised but her position is completely and utterly untenable and she needs to resign."