The Dutchman had been awarded a foul with a quarter of an hour to go by referee Michael Oliver, only to have the ball kicked at his head with force from the Wales national team captain, an incident that Sir Alex Ferguson later claimed could have killed Van Persie.
However, speaking after the game, Williams denied all suggestions that he had deliberately intended to hurt the Netherlands international.
"I've seen it on the TV and that's his [Ferguson's] opinion," the 28-year-old told reporters.
"Everyone's going to have their own opinion but, from my point of view, I tried to apologise on the pitch but it all flared up.
"I just kicked the ball in frustration and obviously not trying to hit him square on the head. I understand exactly why he's angry. I'd be the same if the ball hit me on the head at that pace."
Williams would later reiterate that point on his own official twitter account, staying: "Obviously the RVP incident was unintentional #OverReaction. Fair play to @WayneRooney for stating the obvious!"
The reference to the Manchester United striker comes after Rooney defended Williams to reporters post-match.
"I think it's one of those things," said the England international.
"The whistle's gone, the defender has gone to clear and it's hit him in the head. I think probably the right decision from the referee."