WASHINGTON—The Supreme Court ruled that bedrock federal civil rights law prohibits employers from discriminating against workers on the basis of their sexual orientation and gender identity, a decision that cements new nationwide protections for LGBT employees.

The high court, in a 6-3 decision, said the broad language of the Civil Rights Act of 1964, which outlaws workplace discrimination on the basis of sex, should be read to cover sexual orientation as well. Conservative Justice Neil Gorsuch wrote the opinion, which was…

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