The statement was circulated to the 15-member council on Wednesday during an emergency meeting on Aleppo but was dismissed by Russian Ambassador Vitaly Churkin as a "propaganda coup".
British Ambassador Matthew Rycroft said Russia's refusal to back the statement "speaks volumes about their support for and protection of the Assad regime."
President Bashar al-Assad's forces launched an offensive in Aleppo on April 22 that they said was aimed at flushing out jihadists. But the West has accused Damascus of targeting civilians, hitting hospitals and markets.
After air strikes hit a camp of displaced Syrians near the Turkish border on Thursday, the British ambassador called for more pressure to be exerted to rein in the Damascus regime.
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"It is really high time for every member of the Security Council to use every last drop of influence on the Assad regime to get them to respect the cessation of hostilities and every other obligation under international humanitarian law," said Rycroft.
At least 28 civilians, including women and children, were killed Thursday in the bombing of the camp in Sarmada in Idlib province, which is controlled by Syria's Al-Qaeda-linked Al-Nusra Front.
The proposed statement would have condemned the upsurge in Aleppo and attribute the violence to the Syrian military offensive.
Addressing the council on Wednesday, Syria's Deputy UN Ambassador Mounzer Mounzer insisted that regime forces were taking on terror groups in Aleppo.
"What the Syrian government has been doing in the city of Aleppo is merely the fulfilment of its obligations to protect its citizens from terrorism," he said.