The surrogacy arrangement between West, a nurse based in Alaska, and Ahmed and Gilkar, of California, was upended about four months into the pregnancy when an ultrasound revealed the baby had hypoplastic left heart syndrome, a congenital condition in which the left side of the heart fails to form properly during fetal development.

The couple asked West to terminate the pregnancy after learning of the diagnosis, but she continued it. She then traveled to Texas, where she gave birth and where the newborn began receiving treatment for the heart condition.

A California court ruled that Ahmed and Gilkar are the child’s legal parents. A Texas court, where West and the baby appeared during treatment, has also affirmed that the couple are the legal parents. West has argued that her actions fell outside the terms of her surrogacy contract with the couple.

West asked the Supreme Court to block the California custody order while she pursued her challenge through the federal court system. The Court has historically been reluctant to step into family-law disputes, particularly when they turn on questions of state law rather than federal constitutional questions. Kagan, who handles emergency applications from California, denied the request Tuesday without opinion or explanation.

Texas Attorney General Ken Paxton had intervened on West’s behalf earlier in the case to obtain a court order allowing her and the infant to receive lifesaving medical care. Paxton has since withdrawn from the case.

The two sides have advanced sharply different accounts of the baby’s medical care. West’s attorneys have said the couple is seeking to prevent the infant from receiving life-saving treatment and that he may die without legal intervention. Lawyers for Ahmed and Gilkar told the Supreme Court in a filing that they have ensured the baby receives such care and will continue to do so “for the rest of his life.”