A senior US official has made clear it is likely to be some time before Saddam Hussein is put on trial.
The Iraqi leader had been hiding out in squalid surroundings
He said a decision on how the former Iraqi leader might be prosecuted had to be taken first, and a mountain of evidence sifted through.
UN Secretary General Kofi Annan has called for a fair and open trial for Saddam Hussein, and said that the UN remained opposed to the death penalty.
The UN will meet later on Tuesday to discuss future plans for Iraq.
Security Council members are to examine the accelerated timetable, drawn up by the US coalition and the US-appointed Iraqi Governing Council, on handing over power to a transitional government next June.
Saddam Hussein is now being held for interrogation at an undisclosed location, but the International Committee of the Red Cross says it hopes the US military will allow a visit so that it can check on his conditions.
The US says that Saddam Hussein is being treated according to the Geneva Convention, but has not yet confirmed that he is considered a prisoner of war.
The Washington official, who refused to be named, has defended the decision to show pictures of Saddam Hussein undergoing a medical examination.
He said it was allowed under the Geneva Conventions in order to maintain peace and security, and it was not just an attempt at humiliation.
Mr Annan said that the former Iraqi leader's capture was "a positive development because Saddam Hussein
has cast a rather long shadow over developments".
"With his capture, that shadow has been removed," he added.
Mr Annan said he believed that outside help might be necessary to ensure an effective legal process in any trial.
When Mr Bush was asked what should happen to Saddam Hussein, he said: "There needs to be a public trial and I'm confident it will be done in a fair way."
The deposed Iraqi leader will get the justice he denied to millions, Mr Bush told a news conference in Washington.
Mr Bush did not say whether Saddam Hussein should eventually face execution.
"This is a brutal dictator...But my personal views are not important in this matter...It's going to be up to the Iraqis to make those decisions."
President Bush also said the work of US-led forces in Iraq remained difficult and would require further sacrifice, but Iraq was on the path to
These words echoed those of UK Prime Minister Tony Blair, who warned that what he called "terrorists and Saddam's sympathisers" would continue their operations despite Saddam Hussein's capture.
SADDAM: POSSIBLE CHARGES
Campaign against the Kurds in the 1980s, including the use of poison gas at Halabja
Suppression of Kurdish and Shia revolts after the first Gulf War
Brutality against the Marsh Arabs
Crimes committed during the wars against Iran and Kuwait
Possible involvement in recent attacks on coalition forces and other targets in Iraq
Underlining US and UK warnings, two car bombs exploded in Baghdad on Monday, killing at least eight people.
Hours later, violence erupted in the flashpoint town of Falluja where pro-Saddam demonstrators stormed the regional government offices.
At least one Iraqi gunman was killed, US military sources and local police say.
A member of the IGC who met Saddam Hussein on Sunday following his arrest has said he seemed a demoralised man who swore profusely but showed absolutely no remorse.
"I found a very broken man...His body language showed that he was very miserable," Muwaffaq al-Rubaiye said.
The IGC has said the former leader should be tried inside Iraq by Iraqi judges "under the supervision of international experts".
Mr Rubaiye said punishment could swift, after the IGC takes over the sovereignty of Iraq at the end of June next year.
"We will get sovereignty on 30 June - I can tell you, he could be executed on 1 July," he said.
Mr Rubaiye, a human rights activist imprisoned under Saddam Hussein, said that a special tribunal was almost ready to start considering war crimes and abuses.
"We're almost there. I can tell you, he's going to be the first," he said.