The Northern Ireland secretary says the UK Government is "truly sorry" over the events in Ballymurphy in 1971.
Brandon Lewis made the comments in the House of Commons on Thursday morning.
It comes after a coroner ruled on Tuesday that all 10 victims of the Ballymurphy massacre were "entirely innocent of any wrongdoing” and use of force by the Army was “unjustified".
Brandon Lewis said the Government "profoundly regrets and is truly sorry" for what happened, and said Prime Minister Boris Johnson is "writing personally to the families".
However, following Mr Lewis's statement, the daughter of one of the victims asked why the apology had not been made by Mr Johnson.
"Why could Boris not do it, why could he not? It won't bring my mummy back but at least you would have felt that you were being respected," said Briege Voyle.
Speaking earlier, Brandon Lewis said: "The events of Ballymurphy should never have happened, the families of those who were killed should never have had to experience the grief and trauma of that loss.
"The vast majority of those who served in Northern Ireland did so with great dignity and professionalism, but it is clear that in some cases, the security forces and the Army made terrible errors too.
"There is no doubt that what happened on those awful few days in Ballymurphy also fuelled further violence and escalation, particularly in the early years of the Troubles."
He continued: "The Government profoundly regrets and is truly sorry for these events, and how investigations after these terrible events were handled, and the additional pain that the families have had to endure in their fight to clear the names of their loved ones since they began their campaign almost five decades ago."
Prime Minister Boris Johnson on Wednesday “apologised unreservedly on behalf of the UK Government for the events that took place in Ballymurphy” and the “huge anguish” caused to the families of those killed.
However, instead of a public apology or one made directly to the families of the 10 victims shot dead by the Army in west Belfast in 1971, it was made to the First and deputy First Ministers in a remote meeting.
Families of some of the victims rejected it as a "third party apology" from the Prime Minister.
"It took the moment away," said John Teggart, whose father was killed in Ballymurphy. "It actually ruined the moment of what we have campaigned for all these years.
"It should have been done right - we deserve that. The way he should have done it, contact the families and make the apology in public, in the houses of Parliament."
According to a Downing Street spokesman, Mr Johnson spoke with Arlene Foster and Michelle O’Neill on Wednesday afternoon.
Neither of the Stormont leaders referenced the apology in statements issued following the virtual meeting which focused on coronavirus.
The spokesman for the Prime Minister said that Mr Johnson “said the conclusions of the Ballymurphy Inquest, published yesterday, were deeply sad and that the events of August 1971 were tragic”.
He continued: “The Prime Minister apologised unreservedly on behalf of the UK Government for the events that took place in Ballymurphy and the huge anguish that the lengthy pursuit of truth has caused the families of those killed.”
SDLP MP Colum Eastwood called on Boris Johnson to "come out of hiding" and meet the families of the Ballymurphy victims.
Mr Lewis replied: "What I would say to (Mr Eastwood), as I've outlined already today, that the Prime Minister is contacting the families directly, as well as his public apology on behalf of the State, and his conversation with the First and Deputy First Minister which I joined him for yesterday."
He added: "But obviously, we will be considering that report in more detail in the period ahead in order to ensure that we are able to reflect properly on that report."
Fresh inquests into the deaths in west Belfast concluded that all 10 victims - including a mother-of-eight and a Catholic priest - were “entirely innocent” and that soldiers were responsible for nine of the fatal shootings.
Coroner Mrs Justice Keegan found that the use of lethal force by the Army was not justified.
She also criticised the lack of investigation into the 10th death, that of John McKerr - a former soldier in the British Army who lost his hand in the Second World War - and said she could not definitively rule who had shot him.
The killings came over three days from 9-11 August 1971 following the controversial introduction of internment without trial.
Soldiers were met with violence across Northern Ireland as they detained IRA suspects.
Misinformation had circulated in the years since that those shot dead in Ballymurphy had been terrorists, but their names have now been cleared - 50 years on.
A solicitor who represents the victims' families has confirmed they have instigated civil proceedings against the Ministry of Defence.
“In light of these findings and the strong criticisms, they will be pushing on with that,” Padraig O Muirigh said.