Mexico's coach Miguel Herrera launched a furious attack on match referee Pedro Proenca for giving the injury time penalty that Jan-Klaas Huntelaar hit home to seal a 2-1 comeback victory.
Robben was at the centre of both incidents in the World Cup last-16 clash in Fortaleza.
"I really have to say and at the same time apologise in the first half I took a dive and I really shouldn't do that," Robben told Dutch broadcaster NOS as he celebrated the victory.
"That was a stupid, stupid thing to do but sometimes you're expecting to be struck and then they pull their leg away at the last minute."
But the Dutch striker insisted Proenca was right to point to the penalty spot after he was challenged by Mexico's veteran captain Rafael Marquez. "I was fouled," he said.
Mexico had been leading until the 88th minute when Wesley Sneijder scored and then Huntelaar put away the penalty.
And Herrera said referee Proenca should take no more part in the World Cup.
"The determining factor was the man with the whistle. He put us of the World Cup," he said.
"Although the first goal was down to our mistake, we had the chance to reconsider during the break before extra time, but at the end when the referee invents a penalty you go out of the World Cup.
"At the very least they can look at this and this gentleman ought to be going home like us."
Herrera claimed his side had been the victim of poor refereeing in three of their four games in Brazil and a European referee should never have been allocated the game in the first place.
"Out of four matches, we had three where the referee was disastrous.
"I don't understand why they had someone from the same confederation. Why not an African, Asian or South American referee All the doubtful decisions went against Mexico."
Giovani dos Santos' strike early in the second-half appeared to have Mexico on course for their first quarter-final appearance at a World Cup on foreign soil.
The Dutch goals stunned the Mexicans however and Herrera admitted his side had lacked experience in how to see the game out.
"Maybe we were missing what teams like Argentina, Uruguay and those with experience of winning do.
"We had four minutes to go plus stoppage time and we should have put the ball in the corner, do what other teams do and perhaps we would have been faced with a different result."
Dutch boss Louis van Gaal admitted that cooling breaks offered to the players for the first time in a World Cup match after half an hour of each half allowed him to have a greater influence on the game.
"It is true it helps having these breaks because first I had switched to 4-3-3 and we created a lot opportunities, (Guillermo) Ochoa had one great save in particular.
"After that I moved to plan B and I did that in the cooling break. That is a clever way of benefiting from these breaks."
The Manchester United manager was vindicated in his substitutions as he surprisingly replaced captain Robin van Persie whilst in need of a goal to bring on Huntelaar.
And van Gaal insisted he was not afraid to remove his star players for the benefit of the team.
"Van Persie's change was a tactical substitution and you have to remember that before the World Cup he was just recovering from an injury so it is difficult for him to keep going.
"I am a coach that wants to win so when I have to I will substitute players."