For Joseph R. Biden Jr. and the Democratic Party that he now leads, the selection of Judge Amy Coney Barrett by President Trump on Saturday puts a name and a face to the urgent threat they believe a 6-3 conservative majority on the Supreme Court poses to decades of progressive laws and protections.

Not that Mr. Biden and the Democrats want to focus on Judge Barrett herself.

Instead, Democrats pressing to slow or stop the advancement of Mr. Trump’s pick are determined to frame the debate around the real-world consequences — especially on health care, but also on abortion — of what would be the biggest ideological swing on the high court since Justice Clarence Thomas replaced Justice Thurgood Marshall a generation ago.

For Mr. Trump, the choice of Judge Barrett represents a welcome chance to shake up the race, in which he has been trailing in the polls, and potentially to address a staggering gender gap by shoring up his standing with conservative women who are alienated by his personal style and other elements of his record.

He and other Republicans hope to personalize the court fight around Judge Barrett, a Catholic with seven children — all of them teenagers or younger — including two adopted from Haiti and a son with Down syndrome.

In the White House Rose Garden ceremony, Mr. Trump described Judge Barrett’s legal qualifications as “unsurpassed” and her record as “beyond reproach.” But he also sought to cast her as a busy working mother with a personal story that, Republicans hope, will resonate with many Americans — and one that will complicate the Democratic calculus about how to respond to her nomination.

“If confirmed, Justice Barrett will make history as the first mother of school-aged children ever to serve on the U.S. Supreme Court,” Mr. Trump said. In her own speech, Judge Barrett talked both about her professional experience and her reputation in her hometown as a “room parent, car pool driver and birthday party planner.”

The nomination also provides Mr. Trump with something of a leading woman as part of his 2020 ticket, and with a ferocious confirmation battle that could shift some national attention away from a pandemic that has cost the lives of more than 200,000 Americans and toward the kind of culture war turf that Mr. Trump is most comfortable occupying.

Nancy Pelosi, the House speaker, signaled immediately that Democrats would oppose the choice from a policy standpoint, not a personal one.

“For four years, President Trump has tried to crush the Affordable Care Act in the Congress and the courts,” she said in a statement. “This nomination threatens the destruction of lifesaving protections for 135 million Americans with pre-existing conditions together with every other benefit and protection of the Affordable Care Act.”

ImageJoseph Biden’s team and other Democrats see the issue of the Supreme Court as a chance to mobilize the Democratic base and win over some swing voters.
Credit…Alex Edelman/Agence France-Presse — Getty Images

A majority of Americans believe the winner of the November election should make the pick, according to recent polling. But in nominating Judge Barrett to replace Justice Ruth Bader Ginsburg, Mr. Trump has pushed to the forefront a complex new stew of factors that could sway or mobilize key voting blocs on both sides of the aisle: Evangelical and conservative Catholic voters, abortion-rights activists and opponents, women and young people.

All of those political crosscurrents will be on display on Tuesday, as Mr. Trump and Mr. Biden head into the first debate of the general election. A sizable share of the waning days of the campaign will be consumed by the court fight; just the hearings and a final vote would account for more than 10 percent of the remaining calendar.

“It’s a little bit of a tricky balance Democrats are going to have to think through,” said Dan Sena, a Democratic strategist who oversaw the party’s takeover of the House of Representatives in 2018.

“Your base is going to want you to stand up and fight for a seat basically viewed as a Democratic-leaning seat,” Mr. Sena added. But depending on how the battle unfolds, he warned, it may “put people in hyperpartisan corners,” which could endanger Democratic gains with swing voters.

Biden campaign officials are wary of any lines of criticism of Judge Barrett’s deeply conservative background that could be perceived as personal attacks.

Mr. Biden’s team continues to cast the race as above all a referendum on Mr. Trump and his stewardship of the pandemic, seeing the issue of the courts chiefly as a chance to mobilize the elements of the Democratic base. For now, the Supreme Court appears unlikely to leapfrog the economy and the pandemic as the central messages in his case for defeating Mr. Trump.

As the president announced his nomination of Judge Barrett, Mr. Biden issued a written statement that focused heavily on the threat to health care and devoted only a few sentences to Judge Barrett directly.

“Even now, in the midst of a global health pandemic, the Trump administration is asking the U.S. Supreme Court to overturn the entire law, including its protections for people with pre-existing conditions,” Mr. Biden said, citing Judge Barrett’s past opposition to the health care law.

Democrats signaled that their strongest political strategy was to focus on the issues at risk with a 6-3 conservative court.

“She’s a woman, but the most important fact is that she’s for getting rid of Roe v. Wade and the Affordable Care Act and has positions that will hurt millions of women,” said Josh Schwerin, a spokesman for Priorities USA, a leading pro-Biden super PAC. “If Republicans think that just by putting up a woman they are checking a box that will help them win back suburban women, they’re wrong.”

Whomever Mr. Trump nominated, Democrats had planned to portray her as a threat to the Affordable Care Act. But Judge Barrett provides particular ammunition after she criticized an opinion by Chief Justice John G. Roberts Jr. preserving the law as having gone “beyond its plausible meaning to save the statute.”

“We’re getting a nominee who is fundamentally at odds with the public on two very core issues,” former Mayor Rahm Emanuel of Chicago said, describing coverage for people with pre-existing conditions and protecting Roe v. Wade as powerful Democratic arguments. “That’s where the line of inquiry should be.”

Senator Kamala Harris, Mr. Biden’s running mate, sits on the Judiciary Committee, providing the Biden campaign with an unusually direct role in the unfolding process. A former prosecutor, Ms. Harris has displayed an exacting style of questioning that helped make her a star and was part of her appeal in joining the ticket in the first place.

The appointment comes as Republicans are already seeking to cut into the traditional Democratic advantage with Hispanic voters, providing fresh relevancy for appeals to more conservative Hispanic Catholic and evangelical voters on matters of religious liberty.

And Penny Nance, the head of the conservative group Concerned Women for America, said she expected that the choice would help Mr. Trump shore up his standing with Catholic women in critical battleground states, in particular in the industrial Midwest. Mr. Trump won white Catholics in 2016, but he is showing some signs of weakness with those voters this year.

“In states like Pennsylvania, Wisconsin, Ohio, you have a strong Catholic base there, and this nominee is right on target,” said Ms. Nance, who attended the White House announcement. 

Even before Judge Barrett was named, Republicans began accusing Democrats of “bigoted attacks on her Christian faith,” as the former White House press secretary Sarah Huckabee Sanders put it on Twitter, praising the judge as “a working mom with impeccable legal credentials.”

Democrats said they hoped not to fall into that trap.

“Groups like mine are not planning to go near that issue at all,” said Brian Fallon, the executive director of Demand Justice, which is spending millions on television ads in key states highlighting the consequences of a new justice.

The first ad flight, Mr. Fallon said, will ask why Republicans are rushing through this pick while not addressing the economic fallout of the pandemic.

A clear majority of Americans, 57 percent, said the next justice should be selected by the winner of the November election, compared with 38 percent who said Mr. Trump and the current Senate should make the appointment, according to a recent Washington Post-ABC News poll. Even greater shares of independents and women supported waiting.

Some progressives are pushing Mr. Biden to embrace expanding the Supreme Court if Judge Barrett is confirmed. Mondaire Jones, who is expected to be a freshman House Democrat from New York next year, said that he planned to introduce the first legislation to do so and that he hoped Mr. Biden had softened his past opposition.

“Like the rest of the world, waiting to see if he has changed his position,” Mr. Jones said.

The court has already reshaped the fight for the Senate — at least financially — as Democrats saw a sudden infusion of tens of millions of dollars.

Jaime Harrison, the Democratic challenger to Senator Lindsey Graham of South Carolina, the Judiciary Committee chairman, added $15.9 million to his television reservations in the past week, according to Advertising Analytics — more than Mr. Graham spent on his entire 2014 re-election.

Mr. Fallon said that he was focused on what he could control — and that that did not include how Republican senators would ultimately vote. He said he wanted to make the confirmation of Judge Barrett both salient for voters and toxic for the G.O.P.

“If I can keep this move unpopular, I can maximize the chances they feel sufficient pain to move this until after the election,” he said. “I can’t make Mitch McConnell not have 53 votes, but I can try to contribute to this move being so unpopular.”

“Then I have at least improved the chances to win the election,” he said, “and that ain’t nothing.”