Rafiq told MPs earlier this month of the “inhuman” treatment he suffered during his time at Yorkshire, with Vaughan among a number of figures implicated in the case.
Former England captain Vaughan denied claims he said “too many of you lot, we need to do something about it” to Rafiq and three other Asian players before a Yorkshire match in 2009.
He has been left off the BBC team for the upcoming Ashes tour of Australia due to a potential “conflict of interest”.
And in an interview with BBC Breakfast shown on Saturday morning, Vaughan said: “I’m sorry for the hurt (Rafiq’s) gone through. Time I don’t think can ever be a healer in the situation that he’s gone through.
“But hopefully time can be a way of us making sure that Yorkshire County Cricket Club never goes through this situation again and never puts themselves in a position of denial that they treated a player so badly.
“It hurts deeply, hurts me that a player has gone through so much (and) be treated so badly at the club that I love.
“I have to take some responsibility for that because I played for Yorkshire County Cricket Club for 18 years and if in any way shape or form I’m responsible for any of his hurt, I apologise for that.”
Rafiq’s account was backed by former Pakistan bowler Rana Naved-ul-Hasan and England leg-spinner Adil Rashid, who said they heard the comment.
When asked if the trio were lying Vaughan said: “The problem with this situation is that we’ve got too much ‘he said, he said, she said, did they say’ and I think we’ve got to move on from accusations of conversations from many years ago.
“There’s a bigger picture here.”
Vaughan was England skipper from 2003 to 2008 and played his entire domestic career at Yorkshire between 1993 and 2009.
“(The alleged incident) was my last few games and I just remember it clearly that I was proud as punch that we had four Asian players representing Yorkshire County Cricket Club,” he said.
“Nothing but a proud, senior, old pro just about to retire and absolutely delighted that Yorkshire had come so far in my time at the club.”
Vaughan also apologised for a series of historic tweets, including one questioning the lack of English speakers in London and another suggesting England spinner Moeen Ali should ask Muslims he does not know if they are terrorists.
“I apologise deeply to anyone that I’ve offended with those tweets,” Vaughan said.
“Times have moved on and I regret those tweets. We all make mistakes and in my life I’ve made quite a few mistakes on Twitter, I apologise for that.”
Vaughan has accepted the BBC’s decision to exclude him from their Ashes coverage, with England’s first Test in Australia starting on December 8.
“I won’t be doing the Ashes which I understand, the editorial at the moment is all about Azeem Rafiq and racism in the game of cricket. I get that,” said the 47-year-old.
“I just hope in time I get that chance to come back. The one thing I love more than anything since I retired is talking cricket. I love being on Test Match Special and hopefully in time I get that chance to do it again.”