Northern Ireland Secretary Karen Bradley will not step down from following comments made about the victims of violence in the Troubles.
Although she told ITV News there is "no getting away," from comments she made in the Commons suggesting deaths in Northern Ireland caused by soldiers and police were not crimes, she remained adamant she wanted to "deliver" for the families of those who lost their lives during the violence.
Criticism of Bradley's comments have come from both the families of victims and Ireland's government.
Downing Street says the prime minister still has full confidence in Bradley as Secretary of State for Northern Ireland.
Video report by ITV News Political Correspondent Libby Wiener
In her apology, Ms Bradley said her language was "wrong" and "deeply insensitive" to many of those who lost loved ones.
She said: "Yesterday I made comments regarding the actions of soldiers during the Troubles. I want to apologise. I am profoundly sorry for the offence and hurt that my words have caused.
"The language was wrong and even though this was not my intention, it was deeply insensitive to many of those who lost loved ones.
"I know from those families that I have met personally just how raw their pain is and I completely understand why they want to see justice properly delivered. I share that aim and that is why I launched the public consultation on addressing the legacy of the Troubles.
"My position and the position of this Government is clear. We believe fundamentally in the rule of law.
"Where there is any evidence of wrongdoing this should be pursued without fear or favour, whoever the perpetrators might be. That is a principle that underpins our approach to dealing with legacy issues and it is one from which we will not depart."
Northern Ireland’s former police ombudsman Baroness Nuala O’Loan urged the Prime Minister to seek Ms Bradley’s resignation.
Ahead of the apology, Baroness O’Loan said: "Those comments show a complete disregard for the operation of the rule of law.
"Moreover, they also demonstrate a total lack of understanding about Northern Ireland and utter contempt for those who suffered the loss of loved ones."
John Kelly, whose brother was killed on Bloody Sunday in 1972, dismissed Ms Bradley’s apology as "too little, too late".
Mr Kelly told BBC Radio 4’s The World At One: "What she said yesterday was a terrible, terrible statement. The families are very hurt by what she said.
"Her credibility has now gone, her impartiality has now gone. In my view, she should resign."
The Irish premier has said Karen Bradley's comments over deaths caused by soldiers and police during the Troubles were insensitive and wrong.
Leo Varadkar said although he respects Ms Bradley, her comments were wrong considering the ongoing search for answers from some victims' families.
"Legacy issues in Northern Ireland, Britain and Ireland are very difficult," he said.
"I've met families who have lost loved ones during the Troubles and they're still grieving, a lot of them are still hurting and have questions that are unanswered and are seeking justice even today.
"In that context, I think the Secretary of State's comments were insensitive and they were wrong.
"Bear in mind we're talking about the killing of civilians, not combatants, peaceful protesters in Derry on Bloody Sunday, we're talking about Ballymurphy, Kingsmill and Dublin and Monaghan," Mr Varadkar said.
"We need a British government that is at least open to the possibility that these killings of civilians were crimes.
"Indeed, there have been convictions for such killings."
Ms Bradley initially told MPs on Wednesday: "The fewer than 10% (of killings) that were at the hands of the military and police were not crimes.
"They were people acting under orders and under instruction and fulfilling their duties in a dignified and appropriate way."
She later returned to the Commons to say: "The point I was seeking to convey was that the overwhelming majority of those who served carried out their duties with courage, professionalism and integrity and within the law.
"I was not referring to any specific cases but expressing a general view.
"Of course, where there is evidence of wrongdoing, it should always be investigated – whoever is responsible.
"These are of course matters for the police and prosecuting authorities, who are independent of Government."
Meanwhile, the country's chief constable has said Ms Bradley has to account for her own words.
Police Service of Northern Ireland (PSNI) chief, George Hamilton, confirmed that if a police officer or soldier shot someone they should be investigated in compliance with the law.
He said: "It is up to individuals to give account for their own words."She has to explain and own her words in the same way as all political leaders do," he added.
He addressed a meeting of his Policing Board oversight body in Belfast.
"Where people have lost their lives we should all be equal under the law. There should be a thorough and effective investigation."
Next week, prosecutors will announce whether soldiers will face trial for the Bloody Sunday killings of 14 innocent civilians in Londonderry.