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 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.