WASHINGTON—The Supreme Court sidestepped major abortion cases Tuesday, letting stand a lower-court ruling that Indiana can’t ban abortions for the purpose of selecting sex or race, or to avoid having a disabled child. It did, however, allow the state to regulate the disposal of fetal remains.

The court’s unsigned opinion appeared the product of a delicate compromise over perhaps the most divisive issue on the docket—one that has resonated more loudly since the majority’s position on abortion fell into question following the...