A Williams clearance smashed into a prone Van Persie during the 1-1 draw, an incident which led to a scuffle and a booking for the Swansea man.
And Manchester United boss Sir Alex Ferguson said after the game that he was convinced Williams had deliberately hit Van Persie.
The United boss said: "It's a disgraceful act and he (Williams) should be banned for a long, long time".
Asked about Ferguson's reaction, Williams told the Daily Telegraph: "It's his opinion, he's entitled to it. I understand why Van Persie was angry but I've kicked the ball away in frustration, I wasn't trying to hit him in the head.
"I tried to say sorry but he wasn't having it. I haven't spoken to him since the game and I didn't shake hands afterwards, but only because I didn't see him. As far as I'm concerned, it's no big deal.
"If I had tried to hit him on the head deliberately, I would have missed! It's not like I'm going to shoot and hit him square in the head like that."