Ghana's Supreme Court has the power to overturn the electoral body's results according to the country's constitution. A decision from the court is not expected for at least several weeks, but the party says neither the delay nor Mahama's inauguration will hamper their court case.
"The inauguration has absolutely no impact either on the Supreme Court's proceedings, or its decisions. The Supreme Court has every authority to declare election results invalid, and if it says so, John Mahama will no longer be President," a statement from the National Patriotic Party said.
Yet international observers said Ghana's election was free and fair, and many wonder how massive fraud could have occurred when members of both parties signed off on results at each polling station and at collation centers.
Some local newspapers have mocked Akufo-Addo for refusing to concede and charged the ruling with trying to disturb the peace.
But the opposition says the constitution allows for election disputes and they have not incited violence.
In his speech, watched by his countrymen and a variety of African leaders and diplomats, Mahama did not bring up the court case or the boycott, rather focusing on the future and promising to make Ghana's economic achievements available to all.
"A tremendous amount of work has been done. Nevertheless, there is still a tremendous amount of work that needs to be done," Mahama said. "Change does not happen overnight. It will appear to darkest before the dawn of a new day makes that progress visible."
Associated Press writer Jon Gambrell in Lagos, Nigeria, contributed to this report.