"We are awaiting your response within two weeks of this announcement," said the speaker in the tape, which was dated December 22. He said armed groups would refrain from attacking withdrawing US forces if it is completed within a month.
Iraqi Sunni groups including al-Qaeda announced in October the creation of what they described as an Islamic state in Iraq.
Addressing the United States, al-Baghdadi said: "We are announcing today our orders for you, so obey them before you regret it.
"We order you to withdrew your troops immediately, using troop carriers and aircraft, and taking only your personal weapons. Don't withdraw any heavy weapons. Instead you should hand that and your military bases over to the holy warriors of the Islamic State," he said on the tape.
Al-Baghdadi claimed that Washington had tried to open a channel of negotiation with his group through the Saudi government. But he had rejected the initiative.
"The giant has started to fall. It is looking for an escape and seeking to negotiate with all the other groups and parties," al-Baghdadi said.
"They sent us al-Salool," he said, employing an abusive historical term to refer to the Saudi ruling family.
Al-Baghdadi said he had spurned the offer because "we are not ones to negotiate with those who killed our children".