LONDON—Elon Musk’s Space Exploration Technologies Corp. has lost a spacecraft launch order from a major satellite operator to a European rival after the failure of one of its rockets and delays in resuming flights.

Inmarsat PLC Thursday said it would move the launch of one of its satellites to the Arianespace Ariane 5 rocket. The London-based satellite-services company said it took the action after delays in the launch schedule for the Falcon 9 rocket, made by SpaceX, as the rocket company is called.

SpaceX...