Addressing a gathering of cheering supporters in Boston, he said he wished Obama and his family well.
It was a "time of great challenges for America", Romney said, adding that he prayed that the "president would be successful in guiding our nation".
He also thanked his running mate Paul Ryan. "Besides my wife Ann, Paul is the best choice I ever made," Romney, 65, said.
"She would have made a wonderful first lady," he added.
Romney, who fought a tough fight that went down to the wire, said this was a "critical time" for the nation. We can't afford partisan bickering, the Republican challenger to the White House told the crowds, many of whom appeared to be listening with tears in their eyes.
"I believe in America. I believe in the people of America," he said.
Romney's speech came after the US media projected him as getting about 300 electoral votes, a comfortable victory from the magic figure of 270 needed to win.
His wife, children and grandchildren came up on stage after the speech.