Kavanaugh was quick to weigh in, asking his first question of a lawyer representing a defendant about 20 minutes into the first argument.
Mr Kavanaugh moved to the Supreme Court after spending 12 years as a judge on the influential U.S. Court of Appeals in Washington, building a conservative judicial record.
Justice Kavanaugh also faces more than a dozen judicial misconduct complaints over his public statements as a nominee to the Supreme Court.
Kavanaugh's predecessor, Justice Anthony Kennedy, who retired in June, was a more moderate conservative and sometimes sided with the court's four liberal justices. There were no disruptions by protesters during the oral arguments, with security tight in the courtroom.
A petition to impeach Justice Kavanaugh has more than 150,000 signatures.
During the interview, Clinton was asked whether they would be prepared to talk about her husband's sex scandal that led to his impeachment, and how it compares to the accusations against Trump and Kavanaugh.
Protesters held signs that said, "We will not forget" and "We do not consent", following the acrimonious fight that culminated in Kavanaugh's 50-48 confirmation by the Senate on Saturday. Kavanaugh's confirmation gave the Republican president a political victory ahead of crucial November 6 congressional elections.
"You, sir, under historic scrutiny, were proven innocent", Trump said to Kavanaugh. Next to him will be Justice Elena Kagan, who hired him to teach at Harvard Law when she served as dean.
So far this term, there are no blockbuster cases on the court's docket, but significant issues concerning LGBT rights, the Affordable Care Act, the Deferred Action for Childhood Arrivals program and voting rights issues are percolating through the lower courts.
U.S. Supreme Court Associate Justice Brett Kavanaugh speaks during his ceremonial public swearing-in, in the East Room of the White House in Washington, U.S., October 8, 2018. "On the Supreme Court, I will seek to be a force for stability and unity". Kavanaugh appeared less sympathetic to the defendants than Gorsuch.
Trump sees his success in getting Kavanaugh onto the court - tilting the crucial body to the right for potentially years to come - as one of the major successes of his turbulent two-year administration.
Part of the duties of the junior justice include taking notes at the weekly closed-door conferences, where current appeals are voted on and pending cases are added to the docket. Kavanaugh will also reunite with six of his former clerks on the US Circuit Court of Appeals for the District of Columbia who are now clerking for other justices.
"I guess that's why she lost", Trump said of his 2016 Democratic opponent.
Mr Kavanaugh denied the allegations and during a Senate hearing on September 27 accused Democrats of an "orchestrated political hit".
He later expressed regret over some of his comments. The US Supreme Court is supposed to be a sacred institution impervious to partisan politics but the newly confirmed justice Brett Kavanaugh gives the court its staunchest conservative majority in decades.
On Monday evening Kavanaugh, 53, joined family members, friends and President Trump at the White House for a ceremonial swearing-in ceremony, where the new justice tried to ease the partisan wounds from his confirmation process.