WASHINGTON—Judge Ketanji Brown Jackson faced a second full day of questioning from senators on Wednesday, with some Republicans set to further press the Supreme Court nominee over her sentences for convicted criminals, while Democrats maintain hopes for a bipartisan confirmation vote.

Judiciary Committee Chairman Dick Durbin (D., Ill.) opened the hearing with comments defending Judge Jackson’s record on crime, citing her support from the Fraternal Order of Police and the International Association of Chiefs of Police.

“Law enforcement is on your side because you’ve been on their side at critical moments, and your family has dedicated a big part of their lives to law enforcement and you obviously believe it at your core,” he said. “So the soft on crime charge, which leads all others, falls on its face.”

The hearing then turned to questions from the final two senators on the panel for the first round. They each get 30 minutes, followed by a second round of questions in 20-minute blocks.

On Tuesday, Judge Jackson answered senators’ first round of questions about her background as a public defender, her sentencing decisions in child-pornography cases and her judicial philosophy, among other subjects. After more questions Wednesday, the hearings will host testimony from expert witnesses Thursday.

Also See: Judge Jackson Hearings Live Blog

Sen. John Cornyn (R., Texas) and Mr. Durbin argued Wednesday over whether Judge Jackson had called former President George W. Bush and his defense secretary, Donald Rumsfeld, “war criminals,” extending the dispute into a second day.

Mr. Cornyn had asked Judge Jackson on Tuesday, “Why in the world would you call Secretary of Defense Rumsfeld and George W. Bush war criminals” in habeas petitions she had filed on behalf of Guantanamo detainees she represented when she was a public defender.

Judge Jackson said she didn’t recall saying that, and said that in the filings she and colleagues had been “making allegations to preserve issues on behalf of my clients.”

The petitions, which named Messrs. Bush and Rumsfeld as respondents in their official capacities, argued, among other things, that the U.S. government had sanctioned torture against individuals, which would constitute a war crime.

The U.S. government has said that the treatment of some detainees was torture, and the Supreme Court mentioned that in passing in a recent opinion.

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Later Tuesday, Mr. Durbin told Judge Jackson that he had researched Republicans’ assertions and that “there was no time when you called President Bush or Secretary Rumsfeld a—quote—war criminal—unquote.” She responded: “That was correct.”

Mr. Cornyn complained Wednesday that Mr. Durbin’s rebuttal had misrepresented his position and the facts.

“I asked her whether she had called him a war criminal and she said under oath to you, ‘No I did not,’ although the record is plain as can be that she accused him of war crimes,” Mr. Cornyn said, adding that Mr. Durbin shouldn’t be rebutting his points without giving him a chance to respond promptly.

“As I noted yesterday. … These charges don’t hold up,” Mr. Durbin said Wednesday. He asked Judge Jackson to respond.

“Public defenders don’t choose their clients, and yet they have to provide vigorous advocacy,” the judge said. “And as an appellate lawyer, it was my obligation to file habeas petitions on behalf of my clients.”

Some Republican senators have zeroed in on sentences Judge Jackson imposed in child-pornography cases, arguing that she had been too lenient, and those sentencing decisions promised to be a further focus Wednesday.

In one tense back-and-forth about sentencing guidelines Tuesday, Sen. Josh Hawley (R., Mo.) told Judge Jackson, “I’ll be direct with you, I am questioning your discretion and your judgment.”

She defended her record, saying that if stacked next to other judges’ records, it would show “a very similar exercise of attempting to do what it is that judges do—attempting to take into account all of the relevant factors and do justice individually in each case.”

A report published by the U.S. Sentencing Commission last year found that federal judges across the country typically issue sentences below federal guidelines in cases where the offender is not producing child pornography, but is instead possessing, receiving or distributing it.

According to data from the 2019 fiscal year, such offenders received below-guideline sentences in roughly two-thirds of cases. On average, judges issued prison terms at least two years shorter than the minimum term called for by the guidelines. Still, the average prison term for offenders is greater than it was more than a decade ago.

While the hearing Tuesday was largely courteous and calm, there were some flashpoints. Sen. Ted Cruz (R., Texas), pointing to Judge Jackson’s service on the board of a private school in Washington, cited several race-related books recommended by the school in questioning her stance on social-justice teachings and critical race theory. Judge Jackson professed limited familiarity with the theory and said that as a board member she doesn’t review the curriculum.

Mr. Durbin said late Tuesday that he was pleased with how the questioning has gone. The proceedings so far have marked a contrast with several heated confirmations for former President Donald Trump’s court picks.

Most senators on the committee “were not mean at all. They did their job and did it in a respectful, professional way,” he said, while adding that a few senators raised issues in a way he thought crossed a line.

“But that’s their business,” he said. “They have that opportunity.”

Barring unforeseen developments, Judge Jackson’s confirmation is a virtual certainty. While the Senate is divided 50-50 between the two parties, Democrats hold the majority with Vice President Kamala Harris acting as a tiebreaker. Even if no Republican votes for her, she will be confirmed as long as all senators who caucus with Democrats vote yes, since the threshold is a simple majority.

Senate Judiciary Committee Chairman Richard Durbin (D., Ill.), left, speaks with ranking GOP member Chuck Grassley of Iowa before the start of Wednesday’s hearing.

Photo: Chip Somodevilla/Getty Images

Still, a bipartisan vote would be seen as a symbolic victory for the Biden administration and Senate Democrats. Last year three Republicans joined all 50 Democrats and independents to confirm Judge Jackson to the U.S. Court of Appeals for the District of Columbia Circuit.

One of those three, Sen. Lindsey Graham of South Carolina, again signaled he might be a no vote this time. Mr. Graham said he was angry that President Biden had chosen Judge Jackson instead of Judge Michelle Childs of South Carolina, whom Mr. Graham had publicly championed.

“The fact that so many of these left-wing radical groups that would destroy the law as we know it declared war on Michelle Childs and supported you is problematic for me,” he said, addressing Judge Jackson.

Other Republicans seen as potential votes in favor of Judge Jackson’s confirmation are Sens. Mitt Romney of Utah, Lisa Murkowski of Alaska and Susan Collins of Maine. None of them sits on the Senate Judiciary Committee, and they have dropped few clues as to where they stand. Sens. Murkowski and Collins voted to confirm Judge Jackson to the court of appeals last year.

Supreme Court confirmations have become increasingly partisan and contentious in recent years. In 2020, not one Democrat voted to confirm Justice Amy Coney Barrett, nominated by then-President Trump to replace the late liberal Justice Ruth Bader Ginsburg. One GOP senator, Ms. Collins, joined Democrats in voting against Justice Barrett’s confirmation, citing the proximity of the election. It was the first time since 1869 that a Supreme Court nominee had been confirmed without a single vote from a major minority party.

In addition to being the first Black woman nominated for the Supreme Court, Judge Jackson, if confirmed, would be the fourth woman on the current nine-member court.

She told senators Wednesday that her parents grew up in Florida under lawful segregation, and were not allowed to go to school with white children, and she noted that the contrast between her reality growing up in Florida and her parents’ reality was “like night and day, in terms of the opportunities that were available to me.”

“So what my being here is about, I think, at some level, is the progress we have made in this country in a very short period of time,” Judge Jackson said.

Judge Jackson’s nomination is part of a push by the Biden administration to diversify the federal judiciary. Of the roughly 50 life-tenured judges Mr. Biden has placed on the federal courts so far, three-quarters have been women and two-thirds have been members of a minority group.

—Laura Kusisto and Deanna Paul contributed to this article.

Write to Lindsay Wise at lindsay.wise@wsj.com