A statement from the 15-member council was discussed on Friday and again at the weekend, but Russia fiercely opposed it.
The Lithuanian-drafted text stated that the elections held on Sunday in the Lugansk and Donetsk regions run counter to the provisions of the Minsk peace agreement, reached in September.
"They tried to draft a statement last Friday and Saturday. They drafted this paper, but their will was blocked by Russia," said Ukrainian Ambassador Yuriy Sergeyev.
The Russian mission to the United Nations said the statement was "inadequate," but did not elaborate.
UN Secretary General Ban Ki-moon last week deplored the elections after Russia announced it would recognise the result.
Sergeyev said he was consulting with the Security Council on the next steps, but declined to give details.
Diplomats said the Ukraine crisis will be discussed in the coming weeks, when UN officials present their monthly human rights report to the council.
The war in east Ukraine has raised tensions between Russia and the West at a time when the Security Council is seeking to address other world crises such as the jihadist onslaught in Iraq and the war in Syria.
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