Between the 14th hole and the 15th tee of one of the club’s two courses, Mr. Trump installed a flagpole on a stone pedestal overlooking the Potomac, to which he affixed a plaque purportedly designating “The River of Blood.”

“Many great American soldiers, both of the North and South, died at this spot,” the inscription reads. “The casualties were so great that the water would turn red and thus became known as ‘The River of Blood.’ ”

The inscription, beneath his family crest and above Mr. Trump’s full name, concludes: “It is my great honor to have preserved this important section of the Potomac River!”

Like many of Mr. Trump’s claims, the inscription was evidently not fact-checked.

“No. Uh-uh. No way. Nothing like that ever happened there,” said Richard Gillespie, the executive director of the Mosby Heritage Area Association, a historical preservation and education group devoted to an 1,800-square-mile section of the Northern Virginia Piedmont, including the Lowes Island site.

“The only thing that was remotely close to that,” Mr. Gillespie said, was 11 miles up the river at the Battle of Ball’s Bluff in 1861, a rout of Union forces in which several hundred were killed. “The River of Blood?” he added. “Nope, not there.