Judge Kavanaugh has said his upbringing by a public-school teacher imparted lessons about racial and economic inequality, and inspired him to embrace parents’ rights to choose schools for their children. As a young boy, Mr. McGuire recalled using his grandmother’s address to attend a better school outside his neighborhood.

But as a black man leading a majority black school, Mr. McGuire also has been fighting to safeguard students’ civil rights, which he believes Mr. Trump seeks to eradicate. And Judge Kavanaugh’s record gives him pause on that front. The nominee has argued against race-conscious voting policies, using the same arguments that the Trump administration has used to undo policies aimed to fight systemic discrimination. In 1999, Mr. Kavanaugh was a co-author on an amicus brief in support of the Center for Equal Opportunity, a group that opposes affirmative action, with Robert H. Bork, who was openly critical of the 1964 Civil Rights Act.

“I’m representing a school that’s trying to bring hope to a community,” Mr. McGuire said. “And there’s a Supreme Court nominee who tugs at the heart strings of what I’m trying to do who has aligned himself with individuals, and a president, who are trying to wipe out the rights of whole demographics of folks. It puts you between a rock and a hard place.”

There are no voucher cases pending before the Supreme Court, but relevant cases are moving through lower courts: a case in New Mexico, which the Supreme Court kicked back for reconsideration after Trinity, involves the exclusion of religious and private schools from a state textbook-lending program.

Just last week, the Puerto Rico Supreme Court ruled in favor of a government-sponsored voucher program, rejecting an argument from the teachers union that the program violated a “support clause” in the island’s Constitution that prohibits support of nonpublic schools.

The conservative group that represented the families in Puerto Rico, the Institute for Justice, has litigated more than 20 educational choice cases, and has another pending before the Montana Supreme Court. In that case, the state Department of Revenue prohibited families from participating in a tax credit scholarship program if they planned to use the scholarship at religious schools.

On Tuesday, the institute used the Trinity case to petition a United States District Court in Washington to challenge the state’s Blaine Amendment. The group is representing a private Christian academy that was excluded as an option for college students who qualify to earn financial aid through the work-study program.