The 767-300ER aircraft was arriving from the Iranian city of Mashhad with 315 people on board.
Aviation officials said the rear right-side landing gear had failed to deploy.
The plane had reportedly been leased out to carry pilgrims to the holy city. All flights to Medina were halted for 24 hours, the airline said.
One eyewitness at Prince Mohammed bin Abdul Aziz International Airport said the plane had made two unsuccessful attempts to land before making the third emergency approach.
General Authority for Civil Aviation spokesman Khaled bin Abdullah Khaybari said the injuries happened as people rushed to leave the plane.
The jet was carrying 299 passengers and 16 crew. Twelve of the injured were taken to hospital for treatment.
State-owned Saudia is the national carrier of Saudi Arabia.
“Saudia expresses its deep sorrow to all the injured passengers as a result of the accident,” the airline said.
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