The demand before the UMP's internal election machinery said the count after the weekend vote did not take into account three overseas French territories.
Fillon, who lost by just 98 votes to former president Nicolas Sarkozy's ally Jean-Francois Copé, also asked UMP heavyweight Alain Juppe, who had been Sarkozy's foreign minister, to take over as the interim head of the party.
Copé said he was opposed to a recount.
"There were results, we must now rally and work together," Cope told journalists.
Fillon's supporters said if the votes in the French territories of New Caledonia, Mayotte and Wallis and Futuma were counted, he would have won a total of 88,004 votes against 87,978 votes for Copé.