Ms Bradley apologised for the “enormous distress” caused by her gaffe and distanced herself from her own words, saying: “I do not believe what I said, that is not my view.”
The cabinet minister has faced calls to quit after she told MPs that killings carried out by the security forces during the Troubles “were not crimes” but actions of people “fulfilling their duties in a dignified and appropriate way”.
Her comments come at a particularly sensitive time, as the families of the Bloody Sunday victims are waiting to hear whether soldiers involved in the killing of civilians in Derry in 1972 will face prosecution.
Ms Bradley later clarified her remarks but was forced to issue a full apology on Thursday amid a furious backlash from politicians and victims’ families.
She said: “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.”
She vowed to try to rebuild the trust of victims and to reassure them of her intent to deliver on stalled mechanisms to address the legacy of Northern Ireland’s past.
But the families of victims of shootings involving the British army in Belfast in 1971 rejected her invitation to meet.
John Teggart, whose father Danny was shot 14 times at Ballymurphy, said she should “do the dignified and appropriate thing” and resign immediately.
“Ballymurphy massacre families have been requesting a meeting with the secretary of state since she took up her position of secretary of state for Northern Ireland,” he said.
“Karen Bradley hasn’t even replied to these requests.”
He added: “We will not meet her, and have one request for Mrs Bradley – and that is for her to resign immediately.”
Downing Street has insisted Theresa May has full confidence in Ms Bradley, who remains a close ally of the prime minister.
But the pressure on the Northern Ireland secretary to quit has shown no sign of abating in recent days.
Ms Bradley previously faced criticism when she admitted, at the time of her appointment, she did not realise nationalists in Northern Ireland did not vote for unionist parties in elections.
Her difficulties come after significant blunders by two other cabinet ministers, on a day of chaos for Ms May’s cabinet.
Amber Rudd, the work and pensions secretary, apologised for referring to the shadow home secretary Diane Abbott as a “coloured woman”, while Commons leader Andrea Leadsom suggested Islamophobia should be treated as a Foreign Office issue.
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