Ten members voted in favor 10, one against and four others abstained.
"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 Zaporozhye regions.
"I would like everyone, including the authorities in Kiev and their real masters in the West, to hear me and remember that the people of [the four territories] are becoming our citizens. Forever," Putin said. "We call on the Kiev regime to immediately cease fire, cease all hostilities - the war it unleashed in 2014 and return to the negotiating table. We are ready for this," Putin said.
The Russian president also took the opportunity to call on Ukraine to respect the choices made by voters in favor of joining the Russian Federation, adding that Moscow would use all means to protect the newly independent territories.
The Kherson, Zaporozhye and the Donbass republics will officially become part of Russia once lawmakers finalize legislation on their incorporation, and is signed by Putin.