The Union of Islamic Courts does not want to impose a Taleban-style Islamic state in Somalia, says their leader.
Sharif Shaikh Ahmed is seen as a relative moderate
Sharif Shaikh Ahmed told the BBC they had no political aims beyond enabling the people to decide their own future.
His militiamen have been seeking to shore up support after taking over the Somali capital from warlords on Monday.
President Bush has expressed concern over the recent unrest and has warned the US will ensure Somalia does not become a safe haven for terrorists.
"Obviously when there's instability anywhere in the world we're concerned. There is instability in Somalia," Mr Bush said on Tuesday.
"First concern of course would be to make sure that Somalia does not become an al-Qaeda safe haven, doesn't become a place from which terrorists plot and plan," he added.
The US was accused of backing the warlords defeated by the Islamists but has neither confirmed nor denied the persistent reports.
Sharia law concerns
Sharif Sheikh Ahmed told the BBC Arabic service that the Islamic Courts were not a movement.
"They are a kind of popular revolution by the Somali people after 16 years of anarchy and killing, plunder and kidnapping," he said.
"This body is not a political one. Rather we want to give power back to the Somali people so it can make its own decisions and decide its own destiny."
The violence in Mogadishu has been the most serious for a decade with some 330 people killed and about 1,500 injured in the past month.
It began when warlords who had divided Mogadishu into fiefdoms united to form the Anti-Terrorism Alliance to tackle the Islamic courts, who they accused of sheltering foreign al-Qaeda militants.
The Islamic courts deny this. They were originally set up in Mogadishu as a grassroots movement by businessmen to establish some law and order in a city without any judicial system.
The Islamic militia warns it is now within 20km (12 miles) of Jowhar, the town where the defeated warlords fled.
However, clan elders have warned the Islamists not to attack Jowhar.
Somalis hope the Islamist victory in Mogadishu will bring peace
A senior Somali MP told the BBC's Network Africa programme that the takeover of Mogadishu could be a step towards ending years of anarchy.
Former Information Minister Mohamud Jama said the Islamic courts might improve the security situation in the capital and make negotiations with the interim government easier.
On Monday, Prime Minister Ali Mohammed Ghedi, whose government wants talks, sacked four Mogadishu-based warlords who had been serving as ministers, as they were losing the battle for the capital.