By Catherine Lucey, Kristina Peterson, Wall street Journal–

President Trump formally nominated Amy Coney Barrett to the Supreme Court on Saturday, kicking off what is expected to be a rapid confirmation process aimed at installing another conservative-leaning judge just weeks before Election Day.

Judge Barrett, 48 years old, is a member of the Chicago-based Seventh U.S. Circuit Court of Appeals and a former law clerk for the late Justice Antonin Scalia. A finalist for a previous Supreme Court opening that went to Justice Brett Kavanaugh, she was seen as a likely choice given her conservative credentials, strong support among Republican senators and the president’s desire to nominate a woman to succeed the late Justice Ruth Bader Ginsburg.

“Today it is my honor to nominate one of our nation’s most brilliant and gifted legal minds to the Supreme Court,” Mr. Trump said in the Rose Garden, with Judge Barrett at his side. “She is a woman of unparalleled achievement, towering intellect, sterling credentials and unyielding loyalty to the Constitution.”

Mr. Trump called the late Justice Ginsburg a “legal giant and a pioneer for women” and said that making his third nomination to the court was a “very proud moment.” Speaking directly to Judge Barrett, he said: “You are very eminently qualified for this job. You are going to be fantastic.”

Mr. Trump also said that if confirmed, Judge Barrett, who appeared with her husband and seven children, would be the first mother of school-age children to serve on the court. To her children he said, “thank you for sharing your incredible mom with our country.”

“This should be a straightforward and prompt confirmation,” Mr. Trump said. He asked Democrats to provide Judge Barrett with “the respectful and dignified hearing that she deserves.”

Judge Barrett said she was “truly humbled by the prospect of serving on the Supreme Court,” adding that “should I be confirmed I will be mindful of who came before me.”

Justice Ginsburg, Judge Barrett said, “not only broke glass ceilings, she smashed them.” She also spoke of the friendship between Justice Ginsburg and Justice Scalia, whom Judge Barrett described as a mentor, adding that she shared Justice Scalia’s philosophy that “a judge must apply the law as written” and that “judges are not policy makers.”

She also spoke warmly about her family, calling her children her “greatest joy” and her husband a source of “unwavering support.”

Mr. Trump had declined to reveal his choice publicly before the announcement, but by Friday he had informed congressional Republicans and allies, many of whom had urged him to choose Judge Barrett, that she was his pick. A military jet made a one-hour flight Saturday morning from Washington to Indiana to pick up Judge Barrett and her family for the White House ceremony, according to people familiar with the planning.

Senate Republicans, who hold a narrow majority, are preparing a fast-paced schedule to confirm the new justice by Nov. 3, a move many GOP lawmakers say is crucial in the case of delayed or disputed election results.

Reaction to the choice quickly fell along partisan lines.

Senate Majority Leader Mitch McConnell (R., Ky.) called Ms. Barrett “an exceptionally impressive jurist and an exceedingly well-qualified nominee to the Supreme Court” who would receive a confirmation vote in the weeks ahead.

Senate Minority Leader Chuck Schumer (D., N.Y.) said he would strongly oppose her nomination and emphasized the role her confirmation could play in rulings on the 2010 health-care law.

“A vote by any Senator for Judge Amy Coney Barrett is a vote to strike down the Affordable Care Act and eliminate protections for millions of Americans with pre-existing conditions,” Mr. Schumer said in a statement Saturday.

Outside conservative groups had already recorded ads and drafted mailers in favor of Judge Barrett ahead of the announcement because the signals were so strong that she would be picked, and liberal groups are now ramping up efforts to oppose her confirmation.

Senate Judiciary Committee Chairman Lindsey Graham said “Judge Barrett is highly qualified in all the areas that matter—character, integrity, intellect, and judicial disposition.” He added that he was “very committed to ensuring that the nominee gets a challenging, fair, and respectful hearing.”

Democratic presidential candidate Joe Biden issued a statement saying the Senate shouldn’t confirm a nominee until after the November elections and describing Judge Barrett as a threat to President Obama’s health-care law.

“She has a written track record of disagreeing with the U.S. Supreme Court’s decision upholding the Affordable Care Act,” Mr. Biden said.

The White House began reaching out to senators to set up meetings, which are expected to begin early next week, aides said Friday. The Judiciary Committee could hold hearings the week of Oct. 10. The committee could potentially approve the nomination by Oct. 22, and a full Senate vote could happen around Oct. 26.

“There’s plenty of time,” Sen. John Kennedy (R., La.), a member of the Judiciary Committee, said this week.

Sen. Dick Durbin (D., Ill.) disagreed. “This Supreme Court nominee may serve on the court for 30 years,” he said on CNN Saturday. “It is nothing short of outrageous that they want to approve her in fewer than 30 days.”

Democrats on the Judiciary Committee are expected to divide over whether to meet with Ms. Barrett, aides said. Some don’t want to participate in a process they view as illegitimate, while others don’t view that as an effective tactic.

“I think that’s not only respectful, it’s important” to have those meetings and attend the hearings, Mr. Durbin said. “I have questions I want answered by a person who wants to serve on the Supreme Court,” he said, raising in particular his concerns over Mr. Trump’s comments about a disputed election.

But Sen. Richard Blumenthal (D., Conn.), a member of the Judiciary Committee, said on Twitter this week that he would “refuse to legitimize this broken, weaponized process by meeting w/any nominee put forward before the inauguration.”

The nominee—Mr. Trump’s third to the court—could shift the balance of power to the right for decades, with possible implications on policy questions including health care and abortion.

Judge Barrett has long been highly regarded by the president and some key allies, said a person familiar with the president’s thinking.

Former White House counsel Don McGahn advocated for her to be nominated to the Seventh Circuit Court of Appeals, which happened in spring of 2017. That fall, Mr. McGahn and Federalist Society President Leonard Leo met with Mr. Trump in the Oval Office and included Judge Barrett’s name on a list of five proposed additions to an updated list of Supreme Court picks.

Mr. Trump was impressed with Judge Barrett when he met with her in 2018 for the vacancy that he ended up giving to Justice Kavanaugh, the person said, but ultimately he decided to hold her in consideration for a future vacancy.

This time around, she was considered the leading prospect from the beginning.

Justice Ruth Bader Ginsburg, an advocate for women’s equality and leader of the court’s liberal wing, was honored in the U.S. Capitol in a private ceremony Friday. Her death from metastatic pancreatic cancer at the age of 87 prompted widespread public mourning.

Democrats contend that the American public should vote before a new justice is nominated, noting that Republicans made the same assertion when they refused to consider President Obama’s nominee, Merrick Garland, after Justice Scalia’s death in 2016. But they have few options given that the GOP appears to have enough votes to move forward.

Republicans say the situation in 2016 was different because control of the Senate and White House was split between the two parties. Only two GOP senators, Susan Collins of Maine and Lisa Murkowski of Alaska, are opposed to holding a vote on the nominee this year. Ms. Collins has said she would vote no on a nominee before the election. Mr. McConnell can lose no more than three Republicans to confirm the nominee, since Vice President Mike Pence can cast a tiebreaking vote.

One of the court’s biggest pending cases, set for argument the week after the election, involves the future of the Affordable Care Act. If a conservative-leaning judge is seated in time for that argument, she would vote on a case that could bring the entire law’s downfall after a lower court found the mandate to carry health insurance unconstitutional because Congress lowered to zero the tax penalty for failing to maintain coverage.

Coming just weeks before final votes are cast, Mr. Trump has seized on the court vacancy during his campaign appearances in an effort to galvanize Republicans and change the subject from what polls have shown is widespread frustration over his handling of the coronavirus pandemic and racial-justice protests.

Unlike the eight justices on the court, Judge Barrett isn’t a graduate of an Ivy League law school. The New Orleans native graduated from the prestigious Catholic Notre Dame Law School. Before clerking for Justice Scalia, she served as a law clerk to Judge Laurence Silberman, a well-known conservative in Washington. After her Scalia clerkship, she spent three years in private practice before returning to Notre Dame, where she still teaches, to launch her academic career.

A Catholic, like several of the Supreme Court justices, Judge Barrett has been a favorite of social conservatives because of her religious faith and their belief that she could give the Supreme Court an all-important fifth vote to overturn or limit Roe v. Wade—the 1973 precedent that established a constitutional right to end a pregnancy—given that Chief Justice John Roberts has signaled he may have reservations about doing so. Earlier this year, the chief justice, joining the court’s four liberals, voted to strike down a Louisiana abortion law, citing precedent.

Judge Barrett’s religious faith prompted questions during her 2017 confirmation hearing for the Seventh Circuit. A handful of Democratic senators, including Sen. Dianne Feinstein of California, asked Judge Barrett—then a nominee—whether she would be able to separate her religious beliefs from her duty of impartiality as a judge. She said she could, adding that it was never appropriate for a judge to impose her personal convictions on the law.

Three Democrats ultimately voted for Ms. Barrett’s confirmation, which came on a 55-43 vote.

Write to Catherine Lucey at catherine.lucey@wsj.com and Kristina Peterson at kristina.peterson@wsj.com