Santos admitted he got it wrong but wanted fans to know he and Van Persie remain close.
He told The Sun: "Robin's departure to United did not turn him into an enemy for me. We remain friends and he talked about exchanging shirts, which is something players do all the time.
"Did I think it would cause such a commotion? Of course I didn't, otherwise I would have told Robin to exchange shirts away from public view.
"That's why I'd like to apologise to supporters who felt aggrieved by it. I just want people to understand he is my friend and I wouldn't treat him differently because of the move to United.
"He approached me at half-time and at the time I did not think it would create such an uproar.
"I would never refrain from saying 'hi' to a good friend. It surprised me that people latched on to a gesture that promotes friendship and fair play.
"Arsene Wenger had asked the supporters to show respect towards Robin, after all."