It is the fifth time Russia has vetoed a UN resolution on Syria during the more than five-year conflict.
The previous four times Russia was backed by China, but on Saturday China abstained from the vote.
Eleven of the 15 council members voted in favour of the draft resolution.
A UN resolution needs nine votes in favour and no vetoes to be adopted. The veto powers are the US, France, Britain, Russia and China.
After Russia’s veto, the council moved to the second vote on the Russian-drafted text, but it failed to gather enough votes to pass.
“It was a day of high drama at the Security Council,” said James Bays, Al Jazeera’s diplomatic editor, reporting from the UN headquarters in New York.
“The Russian ambassador called it a ‘spectacle’. I was talking to him just moments ago as he was leaving. he said no one wins and we need to go back to diplomacy.
“But certainly some ambassadors were pointing the finger at Russia because, of course, the first resolution that was proposed by France, which suggested that all military aircraft over Aleppo should be grounde,d would have gone through if it wasn’t for the Russian veto.”
After Russia’s proposal failed to gather enough votes, Matthew Rycroft, Britain’s ambassador to the UN, asked Russia to “Stop bombing Aleppo now”.
“It [the Russian proposal] failed because it failed to demand an immediate end to the aerial bombardment of Aleppo,” he said.
“It’s a sham. Just as Russia’s hollow commitment to a political process in Syria is a sham. The indiscrimiunate bombing of civilians in Aleppo is sickening and barbaric. Please stop now.”
In response to Rycroft’s comments, Vitaly Churkin, Russia’s ambassador to the UN, said “the UK should stop supporting terrorists instead”.
“Stop supporting all the villans across the world including terrorists,” he said.
Stop interfering in the internal affairs of soverign states. Stop your colonial habits. Leave the world in peace and then maybe, things will improve in many areas and regions of the world.”
In the run-up to the votes, Frank-Walter Steinmeier, Germany’s foreign minister, said the mounting tensions between the US and Russia had created a situation “more dangerous” than the Cold War.
John Kerry, the US secretary of state, has made clear his anger at the Syrian army’s Russian-backed assault on Aleppo, saying that its bombing of civilians could amount to a war crime.
Syria has made significant advances in its renewed two-week-old offensive in Aleppo, seizing territory to the north and pushing back the front line in the city centre.
The front line had remained largely static since the rebels captured the eastern districts in 2012.
Since the government’s offensive began on September 22, a few days after a joint US-Russia-brokered ceasefire collapsed, at least 290 people – mostly civilians – have been killed in rebel-held areas, 57 of them children, says the Syrian Observatory for Human Rights (SOHR) monitoring group.
And 50 civilians, including nine children, have been killed in rebel shelling on government-held areas of the city, according to the SOHR, which relies on a network of sources on the ground.
It said government forces were making further advances on Saturday in the lead-up to the Security Council session.
The SOHR reported heavy air strikes on the rebel-held Fardos and Sukari neighbourhoods.
The assault by Syrian President Bashar al-Assad’s forces has led to a global outcry after air strikes on hospitals and a UN aid convoy.
The offensive has also elicited a warning from Staffan de Mistura, the UN envoy, that eastern Aleppo could be “totally destroyed” by the year’s end.
Council members held negotiations during the past week on the French proposal for an end to the bombing of Aleppo, access for aid deliveries and a ban on military flights over the city.
Churkin said the French measure was “hastily put together” and suggested it was “not designed to make progress … but to cause a Russian veto.
“I cannot possibly see how we can let this resolution pass”.