Besieged by litigation about its travel ban, its immigration agenda, its prohibition on transgender soldiers and numerous potential conflicts of interest, the presidency of Donald J. Trump has seemed at times like a high-stakes spinoff of “The People’s Court” with Mr. Trump in the starring role as defendant in chief.

On Wednesday, yet another courtroom battle promises to pull the White House into the legal spotlight as crucial arguments are heard in New York in a sweeping lawsuit that is challenging the administration’s marijuana policy by seeking to legalize pot under federal law.

When the suit was filed in Federal District Court in Manhattan in July, it appeared to be an intriguing, if limited, effort to help its five named plaintiffs — among them a former professional football player with a business selling pot-based pain relievers and a 12-year-old girl who treats her chronic epilepsy with medical marijuana. But the case was thrust into national relevance last month when Attorney General Jeff Sessions issued an order encouraging prosecutors to aggressively enforce the federal marijuana law, endangering the viability of the multibillion-dollar cannabis industry in states where it is legal.

In its 98-page complaint, the suit presents its case for legalization not only through a host of constitutional arguments, but also by way of a world-historical tour of marijuana use — from its first purported role 10,000 years ago in the production of Taiwanese pottery to the smoking habits of President Barack Obama in his younger days. It points out that the ancient Egyptians used the drug to treat eye sores and hemorrhoids, and Thomas Jefferson puffed it for his migraines. James Madison credited “sweet hemp” for giving him “insight to create a new and democratic nation,” the suit notes.