"Please, please stop the war and prevent other lives being lost. It is illegal, it has to stop. Blair has blood on his hands," Bigley's elder brother Paul said in a statement to anti-Iraq war activists.
Bigley described Ken as his "best pal" and paid tribute to his bravery and courage.
But he said he would never watch the video of his beheading.
"I won't watch that video. Never," he told the Times. "He's gone now and we grieve and we will cry when we bury him."
Liverpool, the native city of Bigley, observed a day of mourning Saturday as his family struggled to come to terms with his execution.
The sense of grief extended to other parts of Britain, including Birmingham where Muslim leaders opened a book of condolence at the local mosque and decried "barbaric and sinful actions" done in the name of their faith.
In Iraq details of the circumstances behind Bigley's killing were emerging.
Insurgents sources revealed British captive Kenneth Bigley escaped briefly from his captors shortly before they beheaded him in Iraq.
They said Bigley managed to get away for about half an hour with the help of one of his captors before he was caught in farmland near the town of Latifiya, southwest of Baghdad.
Bigley was beheaded in the same area soon after his desperate bid for freedom ended on Thursday afternoon, one source said, adding: "He never made it to the main road."
Bigley was believed to have been
held by Zarqawi
The fate of his accomplice was not immediately known. Bigley had allegedly been held by the Tawhid and Jihad group led by Jordanian Abu Musab al-Zarqawi.
He was taken captive in Baghdad three weeks earlier and was in Iraq working for the UAE-based Gulf Services Company on a contract to provide logistical support for a US army base in Iraq.
The insurgent sources said Bigley's killing might also have been precipitated by a major US-Iraqi military sweep in the last few days against guerrillas roaming a triangle of towns, including Latifiya, southwest of Baghdad.
British embassy officials refused to comment.
Blair's government, which for three weeks refused to bargain with Bigley's captors even after they killed his two American colleagues, revealed on Friday that it had opened secret contacts with them in the days before his death.