Ten members voted in favor 10, one against and four others abstained, according to Sputnik.
"The draft resolution has not been adopted owing to the negative vote of a permanent member of the Council," French Ambassador to the UN De Riviere said at a UN Security Council meeting.
France holds the presidency of the Council for September.
Ahead of the vote, US Ambassador Linda Thomas-Greenfield noted that in the event of the resolution being vetoed, the matter would be taken to the 193-member General Assembly.
Russian Ambassador to the UN Vassily Nebenzia earlier remarked that the US-Albanian draft resolution demonstrates the West's refusal to engage and cooperate within the Council. He called the draft a "low-grade provocation with a goal that is clear to all."
The Friday vote came hours after Russian President Vladimir Putin held a speech before lawmakers in Moscow on the accession of the Lugansk, Donetsk, Kherson, and Zaporizhzhia regions.
The Kherson, Zaporizhzhia, and the Donbas republics will officially become part of Russia once lawmakers finalize legislation on their incorporation, and is signed by Putin.
RHM/PR