Former RUC Chief Constable Ronnie Flanagan has said he won't resign despite calls from some of the relatives of the Omagh bomb victims.
Sir Ronnie Flanagan said he was sorry there had been no convictions
He said he was "desperately sorry" no-one had been convicted for the 1998 Real IRA attack which killed 29 people.
But Sir Ronnie, now Her Majesty's Chief Inspector of Constabulary, said he didn't see what positive outcome his resignation could have.
Last month Sean Hoey, was acquitted of 58 charges including the Omagh bombing.
In his ruling, Lord Justice Weir was scathing in his criticism of the police investigation into the bombing.
The former RUC chief constable said he was desperately sorry but he would not leave his current post.
"I have thought very carefully. The families' thoughts are very important to me," he said.
"But I do not see what positive outcome there would be through my resignation."
Victor Barker, whose son was killed in Omagh, had called for his resignation in the light of the court ruling.
Twenty-nine people were killed in the Omagh bomb
Sir Ronnie made his apology after meeting Mr Barker on Wednesday evening.
Michael Gallagher, whose son Aidan was also killed in the bombing, welcomed the apology but said a public inquiry was needed.
"This is a nine and a half year investigation. There have been tremendous failings right through that period, and they are not just confined to Northern Ireland, because that is only part of the story.
"That's why we feel that some over-arching inquiry (is needed), a public inquiry, that will look at both sides of the border, and find out why this crime was not solved."