US defeated in Iraq
He also criticised militant groups, including Egypt's banned Muslim Brotherhood, for believing in Western-style democracy and taking part in elections.
"Bush, you must confess that you have been defeated in Iraq and in Afghanistan and you will be in Palestine soon," he said in the video broadcast by Al Jazeera television.
For all latest news, follow The Daily Star's Google News channel. Jazeera said the video by Osama bin Laden's second-in-command, which had English subtitles, carried the date of the Muslim lunar month which ended in December.
"I congratulate the Muslims on Islam's victory in Iraq. I said more than a year ago that the Americans' departure from Iraq is only a matter of time," said the bespectacled Zawahri, who wore a white turban and sat next to an assault rifle.
"But they are justifying their withdrawal by saying that the Iraqi forces have reached a satisfactory level."
Bush's critics have demanded an exit strategy for a withdrawal from Iraq, where around 2,200 American military personnel have been killed since the 2003 US-led invasion.
Bush, who has refused to set a timetable for any withdrawal, said on Wednesday reducing troops in Iraq was possible in 2006 but that it would be based on the situation on the ground and decisions by military commanders, not on politics.
Zawahri called on Muslim charities to directly help victims of last year's earthquake in Pakistan and not through Islamabad's government, which he accused of corruption.
In Washington, a US counter-terrorism official said: "US intelligence is assessing the video tape. It appears to be a recitation of well-worn jihadi themes. There's nothing particularly new."
Egyptian-born Zawahri said elections in his home country were a "US game ... designed to mislead the Muslim public."
"America tells Islamists: 'every time your behaviour improves, we'll give you more (seats), until you become secularists with only a false affiliation to Islam'."
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