The Brazil international came under fire for approaching the Dutchman, who angered supporters with a 24 million pound switch to Arsenal's Premier League rival in the summer, with his side 1-0 behind at the time thanks to the ex-Gunners star's early goal.
"I'd like to apologise to supporters who felt aggrieved by it," Santos told The Sun.
"Did I think it would cause such a commotion? Of course I didn't, otherwise I would have told Robin to exchange the shirts away from public view."
Manager Arsene Wenger declared that the left back would never do such a thing again after being baffled by the premature gesture.
"It's interesting because Santos is a player who cares. If you speak with him, he gives an image of himself which is not him," the Gunners boss told the club's official website.
"He is serious, he wants to do well. It is about the result, about the team. [Asking for a shirt at halftime is] a difference of culture.
"I can understand that an English guy doesn't understand it and I'm French and I don't understand it either. He is completely in a different world when he does that.
"But he cares and he understands today what it means and, believe me, he will never do it again. I spoke to him about it but I do not want to make too big a story of it.
"I believe that was not the right thing to do at all, but that is not an explanation of our bad performance on Saturday."