The Championship side were due to kick-off at 4pm British time in Santa Cristina, Italy against the second-tier side but instead fielded two Leeds XI teams.
The Yorkshire club have been conducting pre-season training in Italy and beat minnows FC Gherdeina 16-0 in their first friendly in the north of the country.
And after the no-show, the club said on Twitter: "Our game with Viitorul Constanta has been forced to be called off due to the Romanian side not turning up at the ground.
"United will now field two Leeds XIs that will play against each other in the absence of an opposition for the travelling supporter."
Leeds’ new head coach David Hockaday also addressed the fans who showed up to see the game.
"As of yet the Romanians haven't turned up but we are ready to play. We have officials, match kit and we are here," he said.
"If they don't turn up we will put on a game for the fans because they deserve it. That will be two Leeds XI sides.”
New signings Stuart Taylor, Tommaso Bianchi and goalkeeper Marco Silvestri all featured in the starting line-ups, but new loan signing Souleymane Doukara did not.
Goals from Steve Morison, Rudy Austin and Noel Hunt secured a 3-1 win for the first Leeds XI, with Matt Smith scoring for the other.