Kofi Annan quits as international envoy to Syria. Reuters

International disarray over the bloody crisis in Syria has been starkly underlined when the UN envoy Kofi Annan announced that he was resigning because of the failure of what he said had become a "mission impossible".

The former UN secretary general said it had been a "sacred duty" to take up the position five months ago to try to find a solution to the conflict. But growing militarisation and a lack of unity among world powers had changed the circumstances.

"At a time when we need – when the Syrian people desperately need – action there continues to be finger-pointing and name-calling in the security council," Annan said on Thursday in a sometimes bitter and frustrated statement he made at the UN's Geneva headquarters.

Annan's six-point plan for peace in Syria was already moribund but his dramatic resignation will serve as its death certificate. It leaves the international community without an effective grip on the most violent chapter of the Arab spring, now morphing into a civil war that has already cost an estimated 20,000 lives.

Sluggish and ineffective diplomacy has been outpaced by a fast-moving and increasingly dangerous situation with the current focus on fighting for Aleppo, the country's second city.

Ban ki-Moon, the current UN chief, said he would appoint another envoy when Annan leaves at the end of August. The White House said his resignation showed the failure of Russia and China to act at the UN security council. "President Assad, despite his promise to abide by the Kofi Annan plan, continues to brutally murder his own people," spokesman Jay Carney told reporters aboard Air Force One.

Britain said Annan's scheme remained valid. But it is unclear what any envoy can do without any readiness by Assad's regime or the rebels to negotiate a peaceful transition that would stop the killing.

David Cameron highlighted the difficulty after his talks with the Russian president Vladimir Putin in London, when he called for tougher UN resolutions to pressure Assad over the "appalling bloodshed" in Syria. Russia has already used its security council veto three times to block any UN action and Putin gave no sign he was ready to change position.

Assad, Annan said, would have to leave office "sooner or later" – a position that was not endorsed by the security council at a meeting in Geneva on 30 June.

Syria said it regretted his departure.

Critics had assailed Annan's plan from the start on the grounds that it allowed Assad to pay lip service to diplomacy and haggle over the terms while pursuing a violent crackdown on the opposition.

Apart from a few days in April, the Syrian government ignored calls for a ceasefire and the withdrawal of forces from cities. Few prisoners were released, access for humanitarian workers and the media was limited and political dialogue proved impossible as positions polarised.

UN monitors charged with observing the ceasefire moved agonisingly slowly, taking six weeks to deploy to full strength of 300 men who could report on the aftermath of increasingly frequent massacres but were powerless to stop them.

If the failure was of the mission rather than the man, it will still be a blow for the veteran Ghanaian diplomat who has often been criticised for his role as head of UN peacekeeping operations at the time of the Rwandan genocide in 1994 and the massacre of Bosnian Muslims by Serbs at Srebrenica the following year.

"You have to understand: as an envoy, I can't want peace more than the protagonists, more than the security council or the international community for that matter," he said . "My central concern from the start has been the welfare of the Syrian people. Syria can still be saved from the worst calamity – if the international community can show the courage and leadership necessary to compromise on their partial interests for the sake of the Syrian people."

Ban, he said, might find a replacement. "Let me say that the world is full of crazy people like me, so don't be surprised if someone else decides to take it on."

Annan foreshadowed his resignation in an interview with the Guardian last month when he complained of "destructive competition" between world powers over Syria.