Bakir Izetbegovic, a member of Bosnia's tripartite presidency, said he was seeking "truth and justice".
Bosnian Serb officials warned the move would trigger a "serious crisis" in the country.
In 2007, the International Court of Justice (ICJ) cleared Serbia of direct responsibility for genocide.
The ICJ found only one act of genocide during Bosnia's 1992-95 war - the massacre of about 8,000 Bosnian Muslim (Bosniak) men and boys by Bosnian Serb forces in the town of Srebrenica in 1995.
It was Europe's worst atrocity since World War Two.
The ICJ ruled that Serbia had violated international law by failing to prevent the killings, but absolved it of direct responsibility.
On Thursday, Mr Izetbegovic told reporters that "the request [for a review] is being submitted at this very moment".
"I think that I am on the path of truth and justice," he added.
The appeal was launched before a 10-year deadline expires on 26 February.
Mladen Ivanic, the Serb member of the presidency, said the move should have been made collectively by the tripartite body, where a Bosnian Croat also has a seat.
"I am afraid that we have entered a really serious crisis," Mr Ivanic said.
Meanwhile, Serbia Prime Minister Aleksandar Vucic earlier described Mr Izetbegovic's intentions as "bad" for relations between the two countries.
Following the 1992-1995 war, Bosnia is now an independent state under international administration.
Its three main ethnic groups are Bosniaks (Bosnian Muslims), Croats and Serbs.
Politicians from Bosnia's two semi-independent entities - the Muslim-Croat Federation and the Republika Srpska - have often clashed, triggering political crises.
The powers of the central government in the capital Sarajevo are very limited.

But the committee said imposing financial financial penalties on those who did not meet targets would be counter productive.
Scotland is committed to reducing emissions by at least 80% by 2050.
But a recent report confirmed the country again missed its target for emissions in 2012.
It was the third time an annual target had not been met.
The Scottish Parliament's Rural Affairs, Climate Change and Environment Committee voiced its support for mandatory reporting as a government consultation on climate change closes at the end of this month.
The committee heard evidence from public bodies and found that "everyone stressed how seriously they take their responsibilities both to reduce their carbon footprint, and to report on their climate change obligations".
It said there were different methods of reporting between local councils, voluntary organisations, police and other emergency services.
And it said clarification was needed before mandatory reporting was introduced.
Committee convener Rob Gibson, an SNP MSP, said: "The legislation passed by the Scottish Parliament commits Scotland to reducing emissions by at least 80% by 2050 and is world-leading in terms of ambition.
"However that ambition has to be translated into action. It is now time to activate real change in approach and behaviour across all underperforming sectors of society, and this includes the public sector.
"The committee is encouraged to hear many examples of the positive impact voluntary reporting has had on actually reducing emissions, and by the unanimous support for mandatory climate reporting by the public sector bodies who gave evidence.
"We must support those who are not delivering as much as they can to make Scotland a world leader, and encourage them to engage and improve."
Net emissions in 2012 were 55.67 million metric tons of carbon dioxide equivalent (MtCO2e), compared to the target of 53.226.
Welcoming the committee's recommendation, WWF Scotland director Lang Banks said: "If Scotland is to get itself back on track to meeting its climate change objectives then its important everyone plays their part, and that includes Scotland's public sector bodies.
"With better coordination and support from central government, public sector bodies could be slashing their climate pollution while also saving money through initiatives to improve energy efficiency or by generating their own renewable energy."
Bosnia has formally asked the UN's top court to review its ruling which cleared Serbia of genocide in the 1990s, Bosnia's Muslim leader has said.
[NEXT_CONCEPT]
Public sector bodies in Scotland should be compelled to report climate emissions targets and progress, a Holyrood committee has recommended.