Abadi, named on Monday to form a new government, said that defeating terrorism is his chief priority because it directly threatens the conflict-torn country, Efe reported.
The former deputy speaker called on all political movements to believe in the constitution and in the democratic political process and to sit down together, not to form a government, but to develop a national determination to deal with our problems.
He also asked for the cooperation of allied nations and international organisations to fight the jihadist Islamic State (IS).
Abadi, whose nomination was welcomed by the US and Iranian governments, received congratulations on Tuesday from Saudi Arabia's King Abdullah.
President Fouad Massoum, a Kurd, turned to Abadi a day after Maliki said he was suing the head of state for failing to nominate him for a third term as prime minister.
Maliki's Dawa party got the largest number of seats in April elections, but the broader Shiite coalition of which it is a part, the Iraqi National Alliance, has soured on the premier, whose rule has alienated many Sunnis and Kurds.
Abadi, himself a member of Dawa, extended an olive branch to Maliki Tuesday, praising his efforts to combat terrorism and build the Iraqi state, and assuring him he will continue to be a "brother, comrade and key partner" in the political process.
Maliki Tuesday urged the army and police to stay out of the political crisis and limit themselves to their duties of maintaining security and protecting the country.
Meanwhile, Iraqi forces and Kurdish troops carried on their battle against IS militants Tuesday.
Over 80 IS combatants were killed in Iraqi air strikes.
The air strikes were aimed at "important" IS positions near the northern city of Mosul, the security chief of Nineveh province, Mohammed al-Bayati, told Efe.
The deadliest bombing occurred in the city of Sinjar, recently overrun by the IS.
At the same time, at least two people died, including a pilot, and another 20 were injured when a helicopter crashed that was transporting humanitarian aid to members of the Yazidi religious minority who fled to Sinjar mountain after IS seized the city.
US forces also carried out further air strikes aimed at preventing an IS assault on Erbil, capital of the Kurdish autonomous region.