Under Fiji's constitution, the president holds largely ceremonial powers, including making appointments on the advice of the prime minister.
Bainimarama removed Ratu Josefa Iloilo and his vice-president in the December 5 coup.
Resignation
Also on Thursday, Bainimarama announced the resignation of Jona Senilagakali, the country's caretaker prime minister, whom he appointed last month.
The military leader had appointed Senilagakali after disbanding cabinet, suspending parliament and banishing Laisenia Qarase, the prime minister, to his home island 300km north of the capital, Suva.
Bainimarama says Qarase's government was corrupt and has pledged a "clean up".
But his plans met with resistance from the nation's influential Great Council of Chiefs, which demanded that the ousted leaders be reinstated.
The council proposed that Iloilo appoint an interim government and military forces return to barracks as soon as a national unity government is in place.
But in his first public address since the coup, Iloilo said he would have done "exactly what the army commander did" at the time of the coup, and that it was valid under the law.
"Therefore I fully endorse the actions of the commander and the Republic of Fiji Military Forces in acting in the interest of the nation and in upholding the constitution,'' he said in a national address on Thursday.
He added that Fiji should return to democracy as soon as possible and that he planned to hold consultations before appointing the interim government.