Mr Trump will become the 45th US president after securing a surprise victory over Hillary Clinton.
Mr Obama mounted a fierce campaign to prevent Mr Trump from winning the White House, branding him "unfit" for office.
But the president has now urged all Americans to accept the result of Tuesday's election.
"We are now all rooting for his success in uniting and leading the country," Barack Obama said of Mr Trump.
Despite Mr Obama's calls for unity and Mrs Clinton telling supporters Mr Trump must be given a "chance to lead," his victory has sparked protests in several US states.
Hundreds of Anti-Trump demonstrators in New York staged a march on Trump Tower in Manhattan on Wednesday evening, voicing their opposition to Mr Trump's policies on immigration, gay rights and reproductive rights.
Police earlier erected concrete barriers and other security measures outside the 5th Avenue skyscraper, which is likely to be Mr Trump's headquarters during the transition to power.
There were also reports of protesters blocking the entrance to Trump Towerin Chicago on Wednesday evening, chanting: "No Trump, No KKK, No Fascists USA" and "Not my president!"