"Earlier today (Monday), Vice President (Joe) Biden and I called Dr. Abadi to congratulate him and to urge him to form a new cabinet as quickly as possible -- one that is inclusive of all Iraqis, and one that represents all Iraqis," Xinhua quoted Obama as saying.
"I pledged our support to him," Obama added.
The US president noted that Abadi's nomination is an important step towards forming a new government that can unite Iraq's different communities.
Obama also called on all Iraqi political leaders to work peacefully through the political process in the days ahead.
On Monday, politicians allied to outgoing Iraqi Prime Minister Nuri al-Maliki rejected Abadi, saying he is not representing them and has no legitimacy.
Summing up US military operations in Iraq, Obama said American forces have successfully conducted targeted air strikes to prevent terrorist forces from advancing on the northern city of Erbil and to protect American civilians there over the past few days.
"We have stepped up military advice and assistance to Iraqi and Kurdish forces as they wage the fight against ISIL (Islamic State or IS)," Obama said.
The US has begun shipping weapons to the Kurdish forces battling IS militants, state department spokeswoman Marie Harf said.