Schoolboy Spencer Turner, who won a national competition to design the tribute, was praised by the Duke for “capturing the very essence” of the troops who left their trenches to play football during the First World War.
The bronze memorial at the National Memorial Arboretum in Staffordshire depicts a football encasing a handshake between a German soldier and a member of an English regiment in no man’s land.
Addressing a memorial service attended by FA chairman Greg Dyke and England manager Roy Hodgson, William described the work of remembrance as “stunning”.
He said: “We all grew up with the story of soldiers from both sides putting down their arms to meet in no man’s land on Christmas Day 1914 - when gunfire remarkably gave way to gifts.
“It remains wholly relevant today as a message of hope and humanity, even in the bleakest of times. Football, then as now, had the power to bring people together and break down barriers.”