The Dutchman broke Arsenal fans' hearts by switching allegiances to United eight years ago.
Speaking on The High Performance Podcast, Van Persie said: "It was not only between me and Arsene Wenger, there was more happening with Ivan of course and the way he behaved and the way he handled the whole situation.
"If I look back at that whole situation, I have to look at myself as well, where I could have done better or could have done something differently. If I look back, that open letter I shouldn't have done that.
"If I look back at that, in such a delicate, difficult situation of making a transfer after spending so much time at Arsenal, to make a decision to write an open letter to tell my truth in two pages was impossible.
"I think back then it went to the newspaper because there was not social media. So I did that, and that was because I was disappointed with Ivan and his behaviour.
"But I could not go into detail about what exactly back then, because it's just impossible to tell the whole story behind it. And partly it was the fact that Arsenal did not offer me a new deal. So if I look back at that, I should have done that better."
Asked about the open letter, Van Persie went on to explain his regrets but also singled out Gazidis for criticism.
He added: "And I get that (the criticism). But I can promise you on my kids, if someone comes with proof that Arsenal offered me a deal, I will give you a million [pounds] now, today.
"It was Arsenal's decision not to offer me a deal and that is up to them. After many conversations it became clear that we had different ideas about the club.
"I had seven points where I thought Arsenal could improve and in my opinion those seven points they should start dealing with them straight away to be able to compete with the best teams.
"It doesn't really matter what points they are, what matters is that Ivan decided that he didn't agree with one single point of those seven points – which is fair enough.
"So taking that information on board, Arsenal doesn't offer me a deal, they didn't agree with my views – which were only to help, honest views of how the club should move forward – that is a very clear message."