Al-Zeidi, whose release had been delayed by a day due to "red tape", said he had been beaten, whipped and been given electric shocks during his first few days in custody.
"Muntadhar will go to Greece for medical treatment, because he was injected with unknown chemical drugs and he suffers from a continuous headache," Haidar al-Zeidi, his cousin, said.
The aircraft flying al-Zeidi to Damascus, the Syrian capital, was chartered by the chief executive of his employer, Al-Baghdadiya television.
After being released he was met outside the jail in Baghdad by parliamentarians who support his case, his brother, said.
The authorities had asked his family not to give him a hero's welcome, but people celebrated outside his home in central Baghdad.
Al-Zeidi had initially been sentenced to three years for assaulting a foreign head of state, but had his sentence reduced to a year on appeal.
While throwing his shoes at Bush at a news conference in Baghdad in December 2008, he shouted: "It is the farewell kiss, you dog."
Bush managed to duck the flying size 10 shoes.
Speaking on Tuesday, Al-Zeidi said: "At the time that Prime Minister Nuri al-Maliki said on television that he could not sleep without being reassured on my fate ... I was being tortured in the worst ways, beaten with electric cables and iron bars."
He said his guards had also used simulated drowning - the technique of waterboarding used by the US forces on suspects arrested over the September 11, 2001, attacks.
He told Al Jazeera: "I did this as a revenge for the victims and people killed in Iraq ... now the new US politicians should deal with Arabs properly ... as respectful counterparts not like slaves. I am neither afraid nor regretting what I did."
"Today I am free again but my home is still a prison," he added.
The US has kept its military presence in Iraq for six and half years since it led an invasion to remove Saddam Hussein, the former president.
Al-Baghdadiya showed footage of him arriving at the station wrapped in an Iraqi flag and wearing sunglasses. The staff slaughtered at least three sheep in his honour.
Uday al-Zeidi said: "I wish Bush could see our happiness. When President Bush looks back and turns the pages of his life, he will see the shoes of Muntadhar al-Zeidi on every page."
Mosab Jasim, Al Jazeera's producer in Iraq, said: "I used to go to press conferences with Muntadhar. He's a stable man, not that radical, so I was really surprised that he did that.
"I think that, as he was listening to George Bush and the achievements he said he'd made in Iraq, for an Iraqi man who's been reporting on the suffering of the Iraqi people, I think he just couldn't control himself."
Although Bush laughed off the attack, it caused embarrassment to both him and al-Maliki.
The leaders had been speaking at a joint news conference on Bush's farewell visit to Iraq prior to being succeeded by Barack Obama, then president-elect.
Al-Zeidi faces the prospect of a very different life from his previous existence at Al-Baghdadia, a small, privately owned Cairo-based station, which has continued to pay his salary in jail.
His boss has promised the previously little-known reporter a new home as a reward for loyalty and the publicity that his actions, broadcast live across the world, generated for the station.