Mr Kavanaugh was voted into the Supreme Court on Saturday.
"The Senate confirmation process was contentious and emotional". It capped a fight that seized the national conversation after claims emerged that Kavanaugh had sexually assaulted women three decades ago.
The judge was confirmed by the Senate on Saturday, in a 50-48 vote that largely followed party lines.
Mr Trump initially called her a "compelling" witness, but he later questioned her credibility and mocked her at a rally.
"On behalf of our nation, I want to apologise to Brett and the entire Kavanaugh family for the awful pain and suffering you have been forced to endure", he said at the jurist's White House swearing-in ceremony on Monday.
Mr Trump said every notion of fairness and decency had been broken, and berated the "the awful suffering" the judge had been forced to endure, adding that "under historic scrutiny", he had been "proven innocent".
Media captionTrump: "The main base of the Democrats have shifted so far left we'll end up being Venezuela". "Folks who feel very strongly one way or the other about the issues in front of us should get out and vote and participate". "It was a disgraceful situation brought about by people that are evil".
Prof Ford has been unable to move back home because of "unending" death threats, according to one of her lawyers.
The president - whose Republicans fear losing at least the lower chamber of Congress in November - angrily predicted that the Mr Kavanaugh row would backfire on Democrats. However, Justices Sonia Sotomayor and Elena Kagan did not offer remarks during their respective swearing-in ceremonies, opting, instead, to speak at receptions with then-President Barack Obama regarding their appointments.
After taking the oath of office, Kavanaugh spoke to the audience vowing to serve the law in the Supreme Court.
The clerks are Kim Jackson, who previously worked for Kavanaugh on the federal appeals court in Washington, Shannon Grammel, Megan Lacy and Sara Nommensen.
A handful of Democratic lawmakers, including congressmen Luis Gutierrez of IL and Ted Lieu of California, have pressed for Justice Kavanaugh's removal. Every sitting member of the Supreme Court was present at the ceremony. "I thought the way they conducted themselves, the way they dealt with a high-level, brilliant, going to be a great justice of the Supreme Court - the way they really tortured him and his family - I thought it was a disgrace".