WASHINGTON—The Supreme Court said Tuesday it would consider whether a Colorado website designer had a First Amendment right to refuse to produce same-sex wedding announcements, the latest clash between LGBT and religious rights.

Since its landmark 2015 decision extending marriage rights to same-sex couples, Obergefell v. Hodges, the high court has moved cautiously in carving out constitutional exemptions for organizations and business owners opposed to those rights.

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WASHINGTON—The Supreme Court said Tuesday it would consider whether a Colorado website designer had a First Amendment right to refuse to produce same-sex wedding announcements, the latest clash between LGBT and religious rights.

Since its landmark 2015 decision extending marriage rights to same-sex couples, Obergefell v. Hodges, the high court has moved cautiously in carving out constitutional exemptions for organizations and business owners opposed to those rights.

Last June, the justices decided that a Catholic foster-services agency receiving city funding was entitled to turn away same-sex couples, despite local laws prohibiting such discrimination. The decision was based, however, on the court’s own reading of Philadelphia’s Fair Practices Ordinance and its contract with Catholic Social Services rather than broad constitutional principles.

The court has yet to rule squarely as to whose rights—the LGBT community’s or religious conservatives’—must yield when they directly conflict. The Colorado case granted Tuesday, to be argued in the court’s next term and likely decided by June 2023, suggests the justices may be ready to address that issue.

A Littleton, Colo., company, 303 Creative LLC, and its owner, Lorie Smith, filed suit seeking an exemption from the state Antidiscrimination Act, which requires most shops and other businesses to treat customers equally without regard to factors including race, religion, sex, age, disability and sexual orientation.

Ms. Smith said she plans to begin offering wedding websites but that her religious beliefs preclude her from doing so for same-sex couples. “As a Christian who believes that God gave me the creative gifts that are expressed through this business, I have always strived to honor Him in how I operate it,” she says on 303 Creative’s website. “I am always careful to avoid communicating ideas or messages, or promoting events, products, services, or organizations, that are inconsistent with my religious beliefs.”

In expanding the business to wedding website design, Ms. Smith planned to add to that statement, explaining that she would turn away same-sex couples because preparing their announcements “would compromise my Christian witness and tell a story about marriage that contradicts God’s true story of marriage—the very story He is calling me to promote.”

Federal courts in Colorado, however, found no First Amendment exemption for Ms. Smith, even if other website designers were willing to serve same-sex couples.

Granting 303 Creative an exception “would necessarily relegate LGBT consumers to an inferior market because [Ms. Smith’s] unique services are, by definition, unavailable elsewhere,” a 2-1 panel of the 10th U.S. Circuit Court of Appeals, in Denver, said in July.

“The government doesn’t have the power to silence or compel creative expression under the threat of punishment. It’s shocking that the 10th Circuit would permit Colorado to punish artists whose speech isn’t in line with state-approved ideology,” said Kristen Waggoner, general counsel of Alliance Defending Freedom, a religious advocacy group that represents 303 Creative.

“The U.S. Supreme Court has consistently held that antidiscrimination laws, like Colorado’s, apply to all businesses selling goods and services. Companies cannot turn away LGBT customers just because of who they are,” said Colorado Attorney General Phil Weiser, a Democrat. “We will vigorously defend Colorado’s laws, which protect all Coloradans by preventing discrimination and upholding free speech.”

Federal civil rights law doesn't explicitly protect LGBT people from discrimination, although like Colorado, many state and local ordinances do. Those measures must comply with federal constitutional provisions as the Supreme Court defines them.

In 2020, the Supreme Court for the first time applied sex-discrimination provisions of the federal Civil Rights Act of 1964 to cover LGBT employees. Those cases, however, didn’t involve religious objections to gay and transgender people or their marriages.

Two years earlier, the court sidestepped a similar conflict under the Colorado antidiscrimination law involving a baker who cited religious reasons for refusing to provide cakes for same-sex weddings.

In that case, the court overturned Colorado’s sanctions against the baker after finding some members of a state commission that considered the case made statements suggesting bias against religion.

Write to Jess Bravin at jess.bravin+1@wsj.com