Vitaly Churkin, Russia's ambassador to the UN, confirmed that he had been prepared to use his right of veto had the text been put to a vote.
Washington and its European allies agreed to drop their efforts on Friday and turn the matter over to the six-nation Contact Group [Britain, France, Germany, Italy, Russia and the US].
Zalmay Khalilzad, the US ambassador to the UN, noted that although Russia was also a member of the group, it had no right of veto in the process.
Jean-Marc de la Sabliere, France's ambassador to the UN, said: "We regret that it has been impossible to secure such a resolution in the UN Security Council. We will therefore put on hold discussions of a new resolution.
"We have decided to renew discussions within the Contact Group and with the parties along these lines. The people of Kosovo should be given the opportunity to realise their political and economic development.
"We are determined to assist them in that aim in the period ahead. A timely resolving of Kosovo's status will also enable Serbia to move beyond the conflicts of the 1990s and towards a brighter European and Euro-Atlantic future.
"We believe that resolving Kosovo's status must be achieved as soon as possible."
The French envoy said European Union foreign ministers would discuss Kosovo at a meeting on Monday.
Kosovo has been run by the UN since 1999, after a Nato bombing campaign helped drive out Serb forces carrying out a brutal crackdown on ethnic Albanians, who make up 90 per cent of the population in the province.
The Western compromise draft would have turned over administration of the province from the UN to the EU while maintaining a Nato presence.
Condoleezza Rice, US secretary of state, warned on Thursday that Washington was fully committed to achieving independence for Kosovo, despite Russia's opposition.
Rice did not say whether Washington was prepared to unilaterally recognise an independent Kosovo but said "we will get there one way or another".