The channel has urged the Iraqi government to allow lawyers and the Iraqi Red Crescent to visit him.
The Iraqi military has denied that al-Zaidi has been mistreated while in detention.
'Handed to judiciary'
Al-Zaidi has admitted "aggression against a president" during an appearance before a judge, a judicial spokesman said on Tuesday.
"Al-Zaidi was brought today before the investigating judge in the presence of a defence lawyer and a prosecutor," Abdul Satar Birqadr, a spokesman for Iraq's High Judicial Council, said.
"He admits the action he carried out."
The referral of the case to the Iraqi judiciary is usually the first stage of review proceedings that could lead to a criminal trial being held.
Al-Zaidi will remain in custody until the judge has completed his investigation, Birqadr said.
The court may send him for trial under a clause in the Iraqi penal code that makes it an offence to try to murder Iraqi or foreign presidents.
The sentence could be up to 15 years jail, Birqadr said.
Support for attack
Al-Zaidi's attack on Bush, who ordered the US-led invasion of Iraq in 2003, has been met with broad support across the Arab world.
Iraqis calling for al-Zaidi's release from custody held a second day of protests on Tuesday, with hundreds of students marching in Baghdad.
The demonstrations came a day after thousands of people turned out in Baghdad's Sadr City in a show of support for al-Zaidi.
But the Iraqi government on Monday called al-Zaidi's outburst against Bush a "barbaric and ignominious act".
The outgoing US leader, who was making a surprise visit to Baghdad, had just told reporters that while the war in Iraq was not over "it is decisively on its way to being won," when al-Zaidi hurled abuse - and his footwear - at Bush.
Bush, who had been giving a joint press statement with Nuri al-Maliki, the Iraqi prime minister, ducked behind a podium as the shoes narrowly missed his head.
In Arab culture, showing the soles of the feet is a sign of contempt and the shoe-throwing incident is a reminder of the widespread opposition to the US-led invasion.
Bush's visit to the Iraqi capital came just 37 days before he hands the presidency over to Barack Obama, who has vowed to withdraw troops from Iraq.