With Arsenal trailing 1-0 to the Championship (second division) side, the speedy winger went tumbling after a tackle by Paul Connolly.
"I want to apologise to the managers because I actually dived," Walcott said in a club statement. "I was trying to win the penalty.
"I have heard some players say if there is a slight touch go down and it can work both ways. I am not happy with myself for doing that but I am happy that we got the draw."
Walcott, 21, eventually won a penalty in stoppage time when he was pulled back in the box, allowing captain Cesc Fabregas [ Images ] to equalise and secure a replay at Leeds United's Elland Road.
"I had a little joke with the referee afterwards saying 'that was my first dive, can you tell'?", Walcott added.
"I don't have to own up to it and I can't speak for other players but I have just expressed how I feel. I hope people respect that."