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Justice Elena Kagan on Tuesday refused to intervene in a high-profile custody battle that involved difficult questions surrounding surrogacy and abortion, leaving in place a lower-court order that gave custody to the biological parents in the dispute.

The case drew national attention when a surrogate from Alaska refused to have an abortion after the fetus she was carrying was diagnosed with a serious heart problem. She eventually traveled to Texas and gave birth to the child over the initial objections of the parents, a couple from California.

The baby boy was born in August. He received open-heart surgery and has been in a pediatric intensive care unit at a hospital in Dallas, where the couple have also been living to be with him.

On Tuesday, Justice Kagan, who oversees emergency filings from California, denied the surrogate’s request to block the lower-court ruling in the couple’s favor. She acted on her own without referring the matter to her Supreme Court colleagues. The justice’s order did not include an explanation, as is often the case when the court responds to such filings...