The U.S. Supreme Court is set to hear arguments this Wednesday on the most significant challenge to a law that gives preference to Native American families in foster care and adoption proceedings of Native children.

The outcome could undercut the 1978 Indian Child Welfare Act, which was enacted in response to the alarming rate at which Native American and Alaska Native children were taken from their homes by public and private agencies. Tribes also fear more widespread impacts in the ability to govern themselves if the justices rule against them.

The law requires states to notify tribes and...