US forces had earlier claimed to have killed al-Baghdadi.
Major General William Caldwell, the commander of the multinational force in Iraq, told a press conference in Baghdad that US forces had killed Muharib Abdulatif al-Juburi on May 1.
Brigadier General Abdel Karim Khalaf, operations director at the Iraqi interior minister, told state television that al-Juburi was also known as Abu Omar al-Baghdadi.
"We are giving the Persians, and especially the rulers of Iran, a two-month period to end all kinds of support for the Iraqi Shia government and to stop direct and indirect intervention ... otherwise a severe war is waiting for you," al-Baghdadi said.
Iraq's Shia-led government is backed by the US but closely allied to Iran.
The United States accuses Iran of arming and financing Shia militias in Iraq, charges Tehran denies.
In the recording, al-Baghdadi also gave Sunnis and Arab countries doing business in Iran or with Iranians a two-month deadline to cease their ties.
"We advise and warn every Sunni businessman inside Iran or in Arab countries especially in the Gulf not to take partnership with any Shia Iranian businessman, this is part of the two-month period," he said.
Al-Baghdadi said his group was responsible for two suicide truck bomb attacks in May in Iraq's northern Kurdish region. He said the attacks in Irbil and Makhmur showed the "Islamic jihad" was progressing in the Kurdish areas.