The 18-year-old was replaced by Ryan Jack on the hour mark following a collision with Buddies striker Curtis Main.
Rangers released a statement on Sunday, confirming the player will undergo a “graded return”.
"Leon King suffered a concussion in the second half of yesterday's match and was taken to hospital for assessment,” the club said. “Leon was discharged from hospital late yesterday afternoon with advice.
"Today, Leon is sore but well. He will now rest completely and follow a graded return to play under the supervision of the medical team.”
King is due to feature for Scotland Under-21s this week in a friendly with Iceland at Fir Park. However, due to the concussion protocol his involvement may well be unlikely. Rangers are not in action again until next month due to the break for the World Cup.