Maliki "has become hysterical and has lost his balance," said the statement, published on the Kurdish regional presidency website in English.
Addressing the premier, it continued: "You must apologise to the Iraqi people and step down. You have destroyed the country and someone who has destroyed the country cannot save the country from crises."
The virulent statement came after Maliki charged that the Kurdish regional capital Arbil was harbouring militants fighting his government, including from the Islamic State jihadist group which declared a caliphate in Iraq and Syria last month.
Kurdish troops moved into a swathe of disputed areas after federal security forces withdrew as a jihadist-led alliance of militants swept through northern Iraq last month.
The Kurdish peshmerga fighters were in some places the only rampart against the jihadists, but Barzani has since vowed they would never leave the disputed areas again.