The beheading of British hostage David Haines was in retaliation for David Cameron promising to arm Kurdish fighters in Iraq, says Islamic State.
Aid worker Mr Haines was killed by the insurgent group in a video where he is pictured kneeling in front of an IS extremist who is holding a knife.
The jihadist, who appears to have a British accent, said: "This British man has to pay the price for your promise, Cameron, to arm the Peshmerga against the Islamic State."
He criticised the UK's "evil alliance with America", calling the Prime Minister an "obedient lapdog".
He also said the US "continues to strike the Muslims of Iraq and most recently bombed the Haditha Dam (which) will only accelerate your destruction".
"And playing the role of the obedient lapdog Cameron will only drag you and your people into another bloody and unwinnable war".
The militant added: "If you, Cameron, persist in fighting the Islamic State then you like your master Obama will have the blood of your people on your hands."
Mr Haines, speaking under duress, also talked to the camera, saying he held the Prime Minister "entirely responsible for my execution".
He added: "You entered voluntarily into a coalition with the United States against the Islamic State just as your predecessor Tony Blair did, following a trend against our British prime ministers who can't find the courage to say no to the Americans.
"Unfortunately it is we the British public that in the end will pay the price for our Parliament's selfish decisions."
The video began with an interview clip of the Prime Minister and then featured Mr Haines, dressed in orange overalls, making his remarks.
IS militants previously beheaded two American journalists, James Foley and Steven Sotloff, posting the evidence online in videos featuring a masked jihadist with a British accent.
The Foreign Office said Mr Haines' family wish to be left alone. A spokeswoman said: "If true this is another disgusting murder.
"We are offering the family the family every support possible. They ask to be left alone at this time."