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Walker, signed from Tottenham for £50 million ($64 million) in the close season, was dismissed for two bookings before half-time on his home debut at Eastlands.
The England right-back earned his first yellow card in the 42nd minute for a foul on Leighton Baines.
Just two minutes later, he was booked again as a collision with Dominic Calvert-Lewin triggered his dismissal.
"Gutted with last night," Walker tweeted on Tuesday.
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"I apologise to the fans and my teammates. They did amazingly well to get back into the game and get a point."
City were trailing to Wayne Rooney's 200th Premier League goal when Walker was sent off.
But Pep Guardiola's side managed to escape with a draw thanks to Raheem Sterling's equaliser eight minutes from time.