Conservatives were hoping to get a new justice onto the Supreme Court before a major case involving illegal immigrants’ rights to abortion reaches the justices, but they may end up being disappointed by Judge Brett M. Kavanaugh, President Trump’s nominee to fill the looming vacancy.

Since he participated in the case while on the Circuit Court of Appeals for the District of Columbia, Judge Kavanaugh would have to recuse himself when the case reached the justices, under standard court practice.

The choice ultimately would be up to him, but analysts said they’d be shocked if he didn’t sit out the case, which involves pregnant illegal immigrant teens in government custody who say they have a right to get an abortion in the U.S. and the government must assist them.

“I would expect a Justice Kavanaugh to recuse himself from the class action case should he be confirmed and should it reach the high court,” said Gabe Roth, executive director of the nonpartisan, judicial-accountability group Fix the Court.

Recusals are a tricky part of a judge’s decision-making.

While there are specific standards for district and circuit judges, there aren’t any hard-and-fast rules for the high court — only precedent. Justices set their own standards, and make decisions on a case-by-case basis.

Having a financial interest in a case or having worked on a case in their time before they were on the court are common reasons for recusal.

In another vein, Justice Elena Kagan did not hear cases she had overseen when she was solicitor general under President Obama.

Justices also have traditionally sat out cases they took part in while on lower courts.

Retiring Justice Anthony M. Kennedy sat out case earlier this year after he took part in an earlier iteration when he was an appeals court judge in 1985.

And since he took a seat on the federal bench last year, Justice Neil M. Gorsuch has recused himself in three cases. Two of the three were ones he previously considered while he sat on the 10th U.S. Circuit Court of Appeals.

“The standard is whether their impartiality might reasonably be questioned,” said Elliot Mincberg, senior fellow with People for the American Way.

For Judge Kavanaugh, several of his cases could be heading for the high court.

He dissented from one ruling in 2017 upholding the conviction of Francis Lorenzo, who was deemed guilty of securities fraud because a statement he emailed contained false information. Judge Kavanaugh said Mr. Lorenzo should have been cleared.

Mr. Lorenzo appealed the conviction and the justices have said they will hear the case in their next term, which begins in October.

Adam Feldman, founder of the Empirical SCOTUS blog, said he couldn’t imagine Judge Kavanaugh not recusing himself from Mr. Lorenzo’s case after his prior ruling since most justices refuse to hear cases in which they’ve already issued opinions.

“Lorenzo seems the clearest case of this,” Mr. Feldman said.

Another case headed to the high court out of the Washington circuit involves whether international organizations have the same sort of litigation immunity that foreign governments enjoy.

Judge Kavanaugh wasn’t on the panel that made the original ruling, but he was part of the full court’s decision not to rehear the case at their level.

Luman Mulligan, a law professor at the University of Kansas, said Judge Kavanaugh would likely need to recuse himself in this case too since it would be the safer course to preserve impartiality.

But other experts said the judge could make an argument that he didn’t participate in oral arguments with the original three-judge panel.

“He might feel like he didn’t really sit on the case in the same sense. I think there is some ambiguity about that,” Mr. Mincberg said.

Democrats said Judge Kavanaugh should be prepared to recuse himself from anything related to Mr. Trump’s legal troubles, based on Judge Kavanaugh’s prior academic writings on whether a sitting president can be prosecuted.

“Judge Kavanaugh will have to recuse himself from any issue or case involving the special counsel,” said Sen. Richard Blumenthal, Connecticut Democrat.

But not all legal experts agree with that call.

“My read of his article is that he was talking about whether Congress could pass a statute to postpone any prosecutions. I do not believe he would have to recuse in such a case,” said Josh Blackman, a professor at South Texas College of Law.

However, Mr. Blackman said Judge Kavanaugh will likely have to recuse from the illegal immigrant abortion case if it returns to the high court.

That case involved a teen who jumped the border, was arrested and put into a government-run shelter for Unaccompanied Alien Children (UACs). During her initial medical screening by U.S. authorities she was told she was pregnant, and she asked to have an abortion. The Trump administration balked, saying under federal laws restricting use of taxpayer money it couldn’t facilitate her procedure. Instead, officials sought to quickly place the teen with a sponsor who could then help her with her decision.

A district judge ruled the illegal immigrant teen had an absolute right to an abortion and ordered the government to help her. When the case reached the appeals court Judge Kavanaugh disagreed, ruling the government should be given a chance to find a sponsor. The full appeals court then stepped in and overruled Judge Kavanaugh, and the teen girl was sped to have an abortion before the case could be appealed again.

The Supreme Court weighed in anyway earlier this year, vacating the lower court rulings and sending the matter back for more consideration. The current dispute is over whether other unaccompanied girls are in the same situation as the original plaintiff — a class action — and what should happen in those cases.

That case could wind up back before the high court, and analysts predicted a Justice Kavanaugh would sit it out.

“If the underlying fact scenario is the same, then he would be hard pressed not to recuse,” Mr. Feldman said.

Though the decision to recuse rests entirely with the individual justice, outsiders often pressure them.

The late Justice Antonin Scalia famously rebuked calls in 2004 to recuse himself from a case seeking documents related to Vice President Dick Cheney’s energy task force. The call for recusal came after Scalia took a duck hunting trip with Mr. Cheney.

And Justice Clarence Thomas, who worked as an attorney in the 1970s for the biotech giant Monsanto, declined to recuse himself from the company’s cases in 2009 and 2013.

Advocacy groups had also argued Justice Ruth Bader Ginsburg and Justice Kagan should recuse from the 2015 case over whether to legalize same-sex marriage, because both justices had commented about the issue prior to hearing the case, and even presided over same-sex weddings.

Neither justice ultimately recused herself, and both voted for making same-sex marriage a constitutional right.

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