Prime Minister David Cameron has said there should be no sympathy for "callous murderer" Raoul Moat.
Mr Cameron was speaking after flowers were left at the scene of Moat's death in Northumberland and messages of sympathy to him were left on Facebook.
Pc David Rathband, shot by Moat as he sat in a patrol car, has said he was now blind as a result of the attack.
But he told The Sun newspaper he would not let the incident ruin his life.
Moat is believed to have shot Samantha Stobbart, 22, and killed her boyfriend Chris Brown.
The gunman went on the run and apparently shot himself in Rothbury on Saturday.
Speaking during Prime Minister's Question Time, Mr Cameron said: "It is absolutely clear that Raoul Moat was a callous murderer, full stop, end of story.
"I cannot understand any wave, however small, of public sympathy for this man.
"There should be sympathy for his victims and the havoc he wreaked in that community.
"There should be no sympathy for him."
Conservative MP Chris Heaton-Harris asked him to contact Facebook requesting the RIP Raoul Moat group page, which has attracted more than 30,000 members, be taken down.
He described it as carrying a "whole host of anti-police statements".
Mr Cameron said he was making a "very good point".
But a spokeswoman for Facebook said the page would not be removed on the grounds the site encourages public debate about issues in the media.
She added: "Facebook is a place where people can express their views and discuss things in an open way as they can and do in many other places, and as such we sometimes find people discussing topics others may find distasteful, however that is not a reason in itself to stop a debate from happening.
"The site has measures in place which allow people to report any material they find offensive and each complaint would be investigated on an individual basis."
As well as tributes posted online, flowers and cards have been left outside Moat's home in Fenham, Newcastle, and at the scene of his death in Rothbury.
Meanwhile, Mr Rathband told The Sun newspaper about the moment Moat approached his patrol car, wielding a gun.
"He looked at me and I looked at him and he lifted the gun and discharged the weapon through my window.
"It was a noise I had never heard before but it was a noise that was so painful. My first assumption was it was the noise that was ripping my face off."
Mr Rathband said after a second shot was fired to his shoulder, emergency crews arrived at the scene.
He added: "I asked a paramedic to tell my wife that I loved her and to tell my children that I loved them.
"I knew I was going to die."
Northumbria Police have said they would not comment on the story in the newspaper but that Mr Rathband was interviewed as a victim of a crime, not as a police officer.
On Wednesday, police arrested three men on suspicion of assisting an offender.
A total of 13 people have now been arrested in connection with the Moat inquiry.
Four men and a woman held on suspicion of assisting an offender are currently on police bail.
Two other men have been charged with conspiracy to commit murder and possessing a firearm with intent.
The hunt for Moat began on 3 July after he was suspected of shooting Ms Stobbart and Mr Brown in Birtley, near Gateshead.
Pc Rathband was shot the following day in East Denton as he sat in his patrol car.
After a week on the run, Moat was discovered armed with a shotgun on the banks of the River Coquet at about 1900 BST on Friday.
Armed police cornered him and negotiators were brought in to speak to him.
Everyone has a right to an opinion and Moat was a nutter and a piece of shit. not very often i would agree with Cameron but he has my full support on this one and would have the full support of the majority of decent people in this country. Thankfully there are still more decent people in this world than idiots.
God help us, Come yourself, Don't send Jesus, This is no place for children.
in my opinon, once a person goes off on a gun rampage they have given up their right to live in a civilised society.
The first thing the police should do is take them out as soon as possible, none of this softly softly approach with mediators etc.
Bang....job done.
At least then WE the taxpayers dont have to fund a lifelong laidback lifestyle in prison. These scumbags get more spent on feeding and looking after them inside than most people can earn honestly !
The controversial Facebook tribute page for killer Raoul Moat has been removed from the social network site. The 'RIP Raoul Moat you Legend!' page on the popular internet site was taken offline by its creator, Siobhan O'Dowd.
The Facebook page had attracted over 30,000 supporters since it was set up but the US-based company had refused to remove it.
Ms O'Dowd confirmed she had taken the page down but did not rule out putting it back up.
Asked why she removed the page, she said: "I don't know really. A few of us came to a decision but it's going to be up again running.
"We don't condone what he did, as what he did was wrong.
I feel sorry for the families but he was still a human being at the end of the day."
Ms O'Dowd added: "He had problems and needed help and he didn't get any help."
Asked about the furore over the Facebook group, she said: "To be honest, I didn't think this would be the kind of reaction I would get.
"Everyone is entitled to their own opinion. Even Facebook said that and that's why they wouldn't shut it down."
The page had been condemned by Prime Minister David Cameron, who could not understand sympathy being expressed for Moat, whom he branded a "callous murderer".
But after a Downing Street official telephoned Facebook to raise Mr Cameron's concerns, the company put out a statement last night making clear it had no intention of removing the page.
A Facebook spokeswoman said: "Facebook will remove content that violates our terms when reported to us."
The page's creator, Ms O'Dowd, said she had not been threatened since creating the page.
"I cannot get threatened for having an opinion, surely?"
I have expressed my views on this in another thread and caused a bit of a stir last time so will try again.
For killing the man, shooting his ex and cowardly shooting a copper between the eyes then having another shot I am glad he is dead, it saves the government thousands upon thousands in keeping him locked up.
I have no sympathy for him at all after what he has done, my sympathies go to the parents of the man who died, his ex and the copper and his family.
I do however have sympathy for his friends and family as they had no control over any of this (unless they were shielding him).
Still dont see why we need an investigation costing thousands to find out what happened that night, the man had a gun and died, he chose that route, he had not flipped he was calculated, clever enough to hide for 7 days, clever enough to get food, was aware enough to leave messages and contact the police 50 minutes after shooting the copper so he could boast.
I would however have an investigation into why he was not monitored more after making the threats whilst inside and also why if he asked for help 12 months ago was that npot looked into more seriously.
Facebook, thats tricky, I cant really see a problem for a tribute page but it should be left at that, when people are using it to glorify what he done then it crosses a line in my moral compass.