The move heightened diplomatic tensions ahead of a key Russia-US-UN meeting in Geneva today on organising an international Syria peace conference.
Russian diplomats refused to allow any mention in the statement of President Bashar al-Assad's tactics, diplomats said.
In the face of the obstacles, the United States decided to withdraw the draft which needs the approval of all 15 Security Council members to be released. A spokesman said the US administration was "very disappointed" at the Russian blocking.
The United States wanted the statement to express "outrage at the use of air strikes by the Syrian government, in particular the use of heavy indiscriminate weapons, including Scud missiles and 'barrel bombs,' which were dropped on Aleppo" this week.
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The statement would have expressed concern at the general "escalating level of violence in the Syrian conflict and condemned all violence by all parties."
Russia would not comment publicly on the statement. But it has strongly defended Assad from Security Council action during the 33-month-old war in which the UN says well over 100,000 people have been killed.
"We are very disappointed that a Security Council statement expressing our collective outrage at the brutal and indiscriminant tactics employed by the Syrian regime against civilians has been blocked," said Kurtis Cooper, deputy spokesman for the US mission to the United Nations.
"Regime air raids in and around Aleppo have continued unabated. Surely, at a minimum, the Security Council should be able to condemn such barbarities," said Cooper.
Russia and China have vetoed three UN Security Council resolutions proposed by western nations to increase pressure on Assad. Statements, which are not legally binding, have only rarely been agreed on Syria.