But Sergei Lavrov, Russia's foreign minister, said on Wednesday that his government backed Blair.
Blair's appointment was welcomed by figures in the US, Israel and Palestine.
George Bush, the US president, said: "I am pleased that this capable man has agreed to continue his work for peace in the Middle East."
"Tony will help Palestinians develop the political and economic institutions they will need for a democratic, sovereign state able to provide for its people and live in peace and security with Israel."
Mark Regev, an Israeli foreign ministry spokesman, said: "Tony Blair is a friend of Israel, a friend of the Palestinians and above all a friend of peace.
We are delighted with the idea of working with him."
Mahmoud Abbas, the Palestinian president, expressed satisfaction at the decision.
Saeb Erakat, chief Palestinian negotiator, said: "President Abbas welcomes the nomination of Mr Blair as envoy of the Quartet.
"The president, who was consulted on the matter, has given the assurance that he will work with Mr Blair to arrive at a peaceful solution on the basis of two states."