Al-Zawahiri said in the message: "I ask [Bush], why send 20,000 [troops] only - why not send 50 or 100,000? Aren't you aware that the dogs of Iraq are pining for your troops' dead bodies?
"So send your entire army to be annihilated at the hands of the mujahideen [holy warriors] to free the world from your evil … because Iraq, land of the caliphate and jihad [struggle], is able to bury ten armies like yours, with Allah's help and power."
The message is the first reaction from al-Qaeda's leadership to the new Iraq strategy.
"The US is deluding itself if it thinks it's going to resolve the troubles in Iraq by sending additional troops"
Rnia, Riyadh, Saudi Arabia
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Bush plans to send a further 21,500 US troops to Iraq.
Al-Zawahiri addressed Americans, saying they must "accept the facts of what is happening on the ground, and reject the fantasies with which Bush tries to deceive you".
He said: "You must honestly try to reach a mutual understanding with the Muslims, for then, and only then, you might enjoy security.
"Security is a shared destiny, if we are secure, you might be secure, and if we are safe, you might be safe. And if we are struck and killed, you will definitely - with Allah's permission - be struck and killed. This is the correct equation."
Al-Zawahiri also said that Muslim fighters were in control in Afghanistan.
He said: "Among [Bush's] ravings is that he has deprived al-Qaeda of a safe haven in Afghanistan.
"The entire world bears witness to his naked, barefaced lie, because al-Qaeda and the Taliban under the command of the commander of the faithful Mulla Muhammad Omar are the ones who have deprived America of a safe haven in Afghanistan."
Al-Qaeda's second-in-command also said Islamist fighters would "break the back" of the Ethiopians who got "bogged down" by the US in "a real disaster" in Somalia.
"I announce the good news to Bush: he has bogged down his Ethiopian slaves in a real disaster in Somalia. The mujahidin will break their back."
The Somali government, with the aid of Ethiopian and US forces, recently pushed the Union of Islamic Courts movement out of the capital Mogadishu and southern Somalia.
There was no immediate independent confirmation of the video's authenticity.
The video is the third message by al-Zawahiri in just over a month. In an audiotape posted on the internet on January 5, he urged Somalia's Muslim fighters to carry out suicide attacks on Ethiopian troops fighting in their country.