The US Senate voted 50-48 on Saturday to confirm him, with just one Democrat supporting him.
At a White House ceremony on Monday night, Mr Kavanaugh said he was starting his new job without bitterness.
Trump slammed Kavanaugh's opponents for a "campaign of personal destruction".
The U.S. Court of Appeals for the D.C. Circuit has received complaints about Brett Kavanaugh's statements during hearings on his confirmation to the U.S. Supreme Court, according to a statement Saturday by a judge on the federal appeals court.
President Trump said he was apologizing on behalf of the whole country at a ceremonial White House swearing-in after the Senate confirmed the nomination on Saturday despite sexual assault allegations against Mr Kavanaugh.
The president said Kavanaugh overcame intense scrutiny by Democrats who considered him guilty and didn't want to give him due process.
Retired Justice Anthony Kennedy swears in Brett Kavanaugh at the Supreme Court as his wife Ashley holds the Bible.
The 53-year-old justice told the White House gathering that he would not let the "bitter" confirmation process affect his work on the highest court in the land.
Kavanaugh already has been at the Supreme Court preparing for his first day on the bench Tuesday.
Lee explained that there were many possibilities to explain what the confirmation vote indicated about Kavanaugh's innocence.
Hundreds of protesters had also swarmed the steps of the Supreme Court.
Kavanaugh's nomination was thrown into jeopardy when university research psychologist Christine Blasey Ford came forward with accusations that he had sexually assaulted her when they were teens in the Washington area.
By the way, given the volume and shrillness of anti-Kavanaugh protests - from his original hearings all the way through to his delayed confirmation - this is an interesting detail, isn't it? "It is not a partisan or political institution", he said. That evening, he took his oaths of office in a private ceremony at the Supreme Court while protesters chanted outside the court building. Kavanaugh has also hired four clerks, all women, the first time that has happened. Some Democrats have talked of impeaching Kavanaugh, though the process is hard with a two-thirds Senate majority required to remove a federal judge.
"What we decide is whether the state - which has the burden of proving guilt beyond any reasonable doubt, the highest standard in our legal system - has met that burden of proof", Beskind said.
He thanked the Republicans who stood with Kavanaugh and said "it was a great honor to be involved in this situation".
Senator McConnell called the current partisan divide a "low point", but he blamed Democrats.
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