In court papers, attorney Newton Schwartz Sr. says he filed the action because "the U.S. Constitution is not a popularity document for fair weather only."
"This 229-year question has never been pled, presented to or finally decided by or resolved by the U.S. Supreme Court. Only the U.S. Supreme Court can finally decide, determine judicially and settle this issue now," the lawsuit states.
"However persuasive, one finds each side in this debate, the final decision ultimately rests in the hands of five or more of nine Justices on the Supreme Court as mandated by the Constitution."
Cruz, the son of an American mother and a Cuban father, was born in Canada — a fact that Republican presidential front-runner Donald Trump has gleefully made hay of in recent weeks.
Cruz's campaign insists he is eligible, but the suit argues that a United States president must be a "natural-born citizen" — a term whose definition the Supreme Court hasn't previously considered.
During Thursday's GOP debate, Cruz insisted: "Since September, the Constitution hasn't changed. But the poll numbers have. I recognize that Donald is dismayed that his poll numbers are falling in Iowa. But the facts and the law are really clear."
"Under longstanding U.S. law, the child of a U.S. citizen abroad is a natural born citizen."
Schwartz also argues that if his case is not heard, the question of Cruz's eligibility could erupt into a contested election.