As David Cameron acknowledged that the Isis fighter guilty of the "barbaric and brutal" murder of James Foley appeared to be a British citizen, Obama pledged on Wednesday a relentless response to "cowardly acts of violence".
Shortly after Obama spoke, US central command confirmed 14 new airstrikes against Isis near the Mosul Dam, which the Pentagon and Iraqi forces said on Tuesday was no longer under Isis control. US officials later revealed that American special forces had attempted to rescue Foley and other hostages held by Isis in a recent raid inside Syria. The operation was unsuccessful: no hostages were found at the location targeted by the US team.
Obama's remarks from his holiday in Edgerton, Massachusetts, suggested Washington will refuse to be cowed by militant threats to kill a second American hostage currently being held in Syria. The jihadist featured in the video said a second hostage, believed to be Steven Sotloff, would be the next victim if Obama continued to authorise US air strikes against Isis, also known as Isil.
The president said: "Jim Foley's life stands in stark contrast to his killers. Let's be clear about Isil: Isil speaks for no religion. Their victims are overwhelmingly Muslim and no faith teaches people to massacre innocents. No just god would stand for what they did yesterday and what they do every single day."
John Kerry, the US secretary of state, was even more forthright, saying bluntly that Isis "must be destroyed". In a statement, he said: Make no mistake, we will continue to confront Isil wherever it tried to spread its despicable hatred. The world must know that the United States of America will never back down in the face of such evil."
Earlier, Cameron interrupted a Cornwall holiday to return to No 10 to chair a meeting of the Cobra emergency committee. Ministers agreed to step up Britain's involvement in the fight against Isis by sending military trainers to Baghdad to help the Iraqi security forces.
Guidance to UK police is to be "refreshed" to remind them of their powers to arrest those who incite or support terrorism by issuing leaflets backing Isis. Police will be told that flying flags that "incite or encourage support for the Islamic State" are illegal, a government source said.
The prime minister, who watched the video of the beheading after he returned to Downing Street, said: "Let me condemn completely the barbaric and brutal act that has taken place. And let's be clear what that act is: it is an act of murder. And murder without any act of justification. Now we have not identified the individual responsible on the video. But from what we've seen it looks increasingly likely it is a British citizen. Now this is deeply shocking."
The intelligence agencies, working hard to identify the British citizen featured in the video, will focus on British-born fighters who have joined Isis. It is estimated there are 400 to 500 UK fighters in Syria and Iraq with Isis. It is believed that up to 250 have returned to Britain.
Cameron said: "We know far too many British citizens have travelled to Iraq and travelled to Syria to take part in extremism and violence. And what we must do is to redouble all our efforts to stop people going, to take away the passports of people contemplating travel, to arrest and prosecute those that take part in extremism and violence, to take extremist material off the internet, and do everything we can to keep people safe. And that is what this government will do."
UK government sources said officials were working hard with social media sites to take down the video of the beheading. They said the prime minister had watched it to understand the threat. A government source said: "The prime minister watched that in a contained setting. It was provided for him as a video as part of him informing his understanding of the type of threat we are talking about and the response we need."
The source said Cameron believes Britain and the US will face a lengthy battle against Isis. He said: "Overall, this is going to require a tough, patient, resolute response. It is going to take time. The measures we have got have enabled us to do things."
Obama appealed to allies in the Middle East to join in the battle against Isis. "Governments and peoples across the Middle East, there has to be a common effort to extract this cancer so that it does not spread," he said. He praised Foley as a man who stood for "hope and civility" and said: "All of us feel the ache of his absence. All of us mourn his loss."
An emotional Obama ran through a litany of Isis human rights abuses, from rape to enslavement, calling them "cowardly acts of violence". In a vague reference to Americans held captive by Isis or near its path in Iraq, Obama said the US would "do everything we can to protect our people", a formulation that has preceded US military action in the past.
With the death of the first American in Iraq since the US military withdrew in 2011, pressure is mounting on Obama to expand his already growing and amorphous air war against Isis.
The stated purpose of nearly two weeks of bombing has moved from the rescue of mostly Yazidi Iraqi civilians at risk of genocide to providing air cover for Kurdish and Iraqi forces wresting the strategically vital Mosul Dam away from Isis. Aides to Obama pointed out from the start that threats to critical infrastructure would likely prompt a US reprisal.
Buffeted between calls to destroy Isis and criticism of their shifting rationale, Obama and the Pentagon have strenuously objected to a charge of mission creep. They point out the consistency of the goals Obama has articulated since 8 August: preventing humanitarian catastrophe and protecting US personnel in Iraq.
Yet both are broad enough to encompass aerial protection of Iraqi Kurdistan and the destruction of Isis vehicles, artillery and fixed positions from Mount Sinjar to the Mosul Dam. The murder of Foley – delivered after he gave a statement at knifepoint blaming the US for his death – and Isis's threat to kill another American journalist, Sotloff, is also testing Obama's goal of safeguarding US nationals in a country overrun by Isis.
Cameron said Britain would help with military equipment and is prepared to arm Kurds fighting Isis. But he said Britain would not be sending combat troops.
"I've been very clear this country is not going to get involved in another Iraq war. We're not going to put combat troops, combat boots on the ground – that's not something we should do."