Rumours about the health of Buhari have swirled in the last few months and he is now said to be working from home.
South Africa was one of the countries pushing for the withdrawal.
President Adama Barrow flew out of his tiny west African country on January 13 to meet world leaders in Mali
Former leader leaves after 22 years due to days of intense negotiation, a triumph for African diplomacy.
Barrow gave the oath in a tiny room in Gambia's embassy in the Senegalese capital, Dakar.
Barrow said the inauguration was going to take place as scheduled on Thursday in the Senegalese capital, Dakar.
Nigeria and Ghana committed troops and aircraft to the buildup that’s centered in Senegal, which surrounds Gambia on three sides.
While tensions escalate and Gambians flee to neighbouring Senegal, President Yahya Jammeh has secured another three months in office.
In a televised address, The Gambia's President Yahya Jammeh denounced the 'unprecedented foreign interference' in the country's internal affairs.
The Gambia's top judge said that he could not hear President Yahya Jammeh's inauguration challenge, recusing himself from the hearing.
Regional talks to make incumbent Jammeh step down have been unsuccessful.
On Wednesday, the Gambian press union, one of the most harshly persecuted groups under Jammeh, said he must back down.
Diplomats say if negotiations fail, force may be an option
The Kenyan president has faced charges at the court over his alleged role in the deadly inter-ethnic violence after Kenya's 2007 elections.
Jammeh had conceded defeat on state TV last week, prompting wild celebrations over the defeat of his government.
For 22 years, Jammeh, a former junior army officer, mixed charm and generosity with the threat of violence to maintain a firm grip on power
Voting against Jammeh on Thursday was a rare show of defiance against a leader who human rights groups say routinely crushes dissent.
Gambian opposition leader Adama Barrow said he was still waiting for an official result from the commission but that his own count showed he had won.
The recent withdrawals of South Africa, The Gambia, and Burundi initially sparked fears of a domino effect by other African state parties.