Alaska’s American Civil Liberties Union chapter is suing Gov. Mike Dunleavy after he said he would cut funding for the state’s Supreme Court in retaliation to abortion-related rulings.

In a line-item veto, Mr. Dunleavy cut nearly $335,000 in funding for Alaska’s court system, which the Republican’s office said was the same amount the state pays annually to cover elective abortions under its Medicaid program.

In...