"Reaching a just judgement is not within the domain of the Guardian Council and above all a board which is appointed by this council."
Mousavi, a former prime minister, called for the issue to be referred to a body that had been "approved by all candidates and supported by those top clergy who have supported the resolution of the issue".
Abbas-Ali Kadkhodai, a spokesman for the Guardian Council, told the Mehr news agency on Friday that the candidates had 24 hours to name their representatives to the panel that would recount the votes.
Mohsen Rezai, the former head of the Revolutionary Guard who finished third in the poll, said that he was prepared to sit on the panel, and he called on Mousavi and Karroubi to join him.
The Guardian Council has already stated that it has found no major violations in the vote and the authorities have made it clear that they have no intention of a staging a re-run.
The accusations of vote-rigging prompted a series mass protests by opposition supporters, but they have tailed off after security forces cracked down on demonstrators, using tear gas, water cannon and batons to prevent them from gathering.
Al Jazeera's Alireza Ronaghi, reporting from Tehran, said that despite the rejection of the recount there were no protests planned in the coming days.
"Apparently the crackdown by the police has been so heavy that none of the protesters intend to go out on the streets again and it would be unwise for them to do so," he said.
"Right now what they are planning to do is to continue pressing the authorities for legal permission to hold protests and continue their protests on the rooftops, shouting 'Allahu Akhbar' at ten o'clock every night."
At least 20 people have been killed in the unrest, according to Iran's state media.
Also on Saturday, New York-based Human Rights Watch said pro-government Basij militias were raiding homes in an attempt to stop the protests.
"Witnesses are telling us that the Basijis are trashing entire streets and even neighbourhoods as well as individual homes trying to stop the nightly rooftop protest chants," Sarah Leah Whitson, the group's Middle East director, said in a report.
The Human Rights Watch report quoted one woman as saying that Basijis had climbed over walls to enter homes in Velenjak district on June 23.
"When they entered the homes, they beat the residents," she was quoted as saying.
"The neighbours took to cursing the Basijis and throwing stones at them to divert them from beating the residents, but then the Basijis attacked those neighbours' houses and tried to enter them."
A resident of Vanak district was quoted as saying; "On June 22, while we were shouting 'Allahu Akbar' from the rooftops ... the Basiji entered our neighbourhood and started firing live rounds into the air."
There was no independent confirmation of the claims in the report.