President Barham Saleh says early legislative elections would be in line with the aspirations of the Iraqi people and would guarantee political and social stability.
Rescue teams working through the night were able to provide supplies of oxygen, as well as food and water, to some of those trapped through gaps in the rubble from a Shia Muslim shrine, local media reports.
The postponement exacerbates Iraq's political problems because it is the task of the president to formally name a prime minister, who must be backed by an absolute majority in parliament.
The Supreme Court's decision came after Iraqi MPs submitted a complaint against initially favourite presidential candidate Hoshyar Zebari based on corruption charges.
Iraq's president names a former governor of Najaf city Adnan Zurfi as prime minister-designate, following weeks of political infighting.
In capital Baghdad and southern cities, demonstrators chant "we reject Allawi" and hold posters of the new PM's face with a red cross through it a day after Mohammed Tawfiq Allawi's appointment.
The demonstrations comes ahead a Sunday midnight deadline for naming an interim prime minister. that was missed by the country's parliament.
Iraq's parliament has approved the resignation of Prime Minister Adil Abdul Mahdi on Sunday, amid ongoing violence and anti-government demonstrations in the capital.
Kurdish candidate Barham Saleh sweeps to the post of president of Iraq following which he instructs former vice president and veteran Shia politician Adel Abdel Mahdi to form a government.
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