He said Mr Zaidi added: "But I remember in the summer of 2005, I interviewed your excellency and you told me, 'Come in, this is your house'. And so I appeal to your fatherly feelings to forgive me."
However, according to Reuters news agency, one of Mr Zaidi's brothers expressed scepticism over the merits of the letter.
"This information is absolutely not true. This is a lie. Muntader is my brother and I know him very well. He does not apologise," Udai al-Zaidi said.
He added: "But if it happened, I tell you it happened under pressure."
'Signs of blows'
Judge Dhiya al-Kenani said the shoes at the centre of the incident had been destroyed by US and Iraqi security agents when they were checked for explosives.
"I would have preferred to have had the shoes as evidence for the case but since Muntader al-Zaidi has confessed to his action and that the television pictures confirm it, the investigation can continue," he told the AFP news agency.
President Bush ducks as the shoes are thrown
Judge Dhiya al-Kenani said the court had refused the journalist's request for bail "for the sake of the investigation and for his own security".
He also told the AFP that the journalist had "signs of blows to the face" from when he was arrested, but rejected claims by another of Mr Zaidi's brothers claim that he suffered a broken arm.
Mr Zaidi leapt from his chair at the news conference in Baghdad and hurled first one shoe and then the other at Mr Bush, who was at the podium with Mr Maliki.
The shoes missed as Mr Bush ducked, and Mr Zaidi was wrestled to the ground by security guards and frogmarched from the room.
"This is a farewell kiss, you dog," he yelled in Arabic as he threw his shoes. "This is from the widows, the orphans and those who were killed in Iraq."
Thousands have been protesting in support of his actions.
His employers, the Cairo-based al-Baghdadiya TV channel, called for their journalist to be freed, saying he had simply been exercising his freedom of expression - something the Americans promised Iraqis when they ousted former Iraqi leader Saddam Hussein.
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