The Finucane family have described the British government's decision not to hold at public inquiry into the murder of Pat Finucane as "astonishing, arrogant and cruel".
The family have promised to continue campaigning until their questions were answered.
In a statement the family said: "There is only one reason to ask the local police to investigate a case that involves the British Army, the Security Services and former members of government: it means they will be untouchable.
"It is this internalisation of the issue to Northern Ireland that has allowed those responsible for the murder of Pat Finucane to do so with impunity.
"In failing to establish a public inquiry into the murder of Pat Finucane, the British government have not only set themselves against my family but also the Irish Government, local, national and international political parties, political institutions, legal and human rights groups domestically and internationally."
Pat Finucane's son John said has said his family are very angry and described the decision as nothing short of insulting.
John Finucane said Brandon Lewis's response was a disgrace, adding they had waited 31 years for an effective investigation into the murder of their father.
"We are very angry. What the British government proposed to us today was nothing short of insulting."
"The British government at every opportunity will continue to make the wrong decision and put all their efforts into ensuring that the truth of what happened to my father will not see the light of day and they are intent on suppressing that."
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His brother Michael said his family was "disappointed" but "not surprised" at the decision by the British government.
Speaking following a virtual call with Mr Lewis, Michael said: "I think we were quite angered and exasperated at the Secretary of State's conclusion in the matter because he has proposed that the case is to be resolved by way of an investigation carried out by local police in Northern Ireland.
"I don't think anyone, looking at the evidence in this case and the agencies involved, would propose with any degree of credibility that this could be handled by the local police."
He said the proposal is "inadequate".
"There was a frank exchange of views and he [Brandon Lewis] is in no doubt of the deep dissatisfaction," he added.
"They seem to be relying upon analyses from previous investigations that they have been sitting on for five years."
Taoiseach Micheál Martin said he was disappointed that the British government has not committed to holding a public inquiry, as agreed between the British and Irish governments in 2001.
He said the decision will come as a further blow to Pat Finucane's widow Geraldine and her family, "who have been pursuing truth and justice for three decades with great dignity".
Mr Martin said he noted that the Secretary of State for Northern Ireland did not rule out a public inquiry taking place in the future, and he said the Irish Government remained strongly of the view that a public inquiry was needed.
The Taoiseach noted "that dealing more broadly with legacy issues from the Past was a shared challenge. In this context, delivering on prior commitments, however difficult, was important in terms of building public confidence and trust."
Minister for Foreign Affairs Simon Coveney said: "It has been the strong and consistent position of the Irish Government that only a full and independent public inquiry, as provided for under the Weston Park agreement in 2001, would provide a satisfactory outcome to this case.
"In a case like this, in which Prime Minister Cameron acknowledged that there were 'shocking levels of collusion', there is an undeniable onus on the state to do everything possible to restore public confidence through a process that fully meets relevant international standards and obligations of effectiveness, independence and transparency."
Deputy First Minister, Sinn Féin's Michelle O'Neill said the decision was "shameful" adding that the choice was one of a "rogue state".
Sinn Féin leader Mary Lou McDonald tweeted that "only a full public inquiry can get to the truth. That's why successive British governments have refused to establish one."
DUP MP Jeffrey Donaldson called for a "holistic approach" to legacy that enables all innocent victims to have access to truth and justice.
He said: "We welcome the secretary of state's decision. Equally, we also condemn the murder of Pat Finucane...What we really need is not some special attention to any one case, but a holistic approach to legacy that enables all innocent victims to have access to truth and justice."
SDLP leader Colum Eastwood described the decision as a disgrace.
The decision to reject a public inquiry into the murder of Pat Finucane is a disgrace.— Colum Eastwood (@columeastwood) November 30, 2020
The Secretary of State has sent a very clear message to all victims seeking truth - they won't get it from the British government. pic.twitter.com/XKJnTS6stn
Former Northern Ireland Secretary Theresa Villiers said the appalling events of the Pat Finucane case should not overshadow the work of the majority of those who served in the police and armed services.
She said: "As (Brandon Lewis) said, the murder of Pat Finucane was an appalling and shocking crime, and the involvement of the State as documented by the de Silva review was utterly unacceptable as well.
Shadow Secretary of State for Northern Ireland Louise Haigh said: "I note that he is not ruling a full public inquiry out in the future, why does he not grasp this opportunity to deliver it now?"
"That this crime could happen at all in our country is shocking, that it has never been investigated to a lawful standard is unjustifiable and we have to ask ourselves, as we do with all legacy issues with The Troubles, do we accept a lesser standard of justice for citizens of Northern Ireland than we would if this terrible crime had happened in our constituencies? The secretary of state references the Manchester inquiry, do victims in Northern Ireland not deserve the same transparency and justice?"
The Chairman of the US House of Representatives Ways and Means Committee, Richard Neal, has expressed his disappointment at the British government decision.
Mr Neal co-signed a letter which was sent from members of the US Congress to the UK Prime Minister Boris Johnson last week urging him to establish an inquiry.
"I'm disappointed. I think there was optimistic belief and anticipation that there would be a full inquiry," Mr Neal told RTÉ News.
"All of the evidence at the time pointed to collusion, so I think Geraldine Finucane and her family are entitled to a full review that is not only authentic but stands up under the weight of history," he added.
Mr Neal said he remained hopeful that an inquiry may be held in the future.
"I think that further inspection and more scrutiny can only help the process and bring some peace and solace to the members of the Finucane family," he said.
Democratic Congressman Brendan Boyle has described the decision as "an appalling perversion of justice".
Mr Boyle tweeted that a full public inquiry was long overdue.