Regan today met to discuss the events of the October 17 match with Hugh Dallas, the SFA's head of referee development, as well as the fixture's referee, Dougie McDonald, and Craven, his assistant on the day.
Following the meeting at Hampden Park, Regan said: "I can confirm that I have met with the relevant match officials and members of the referee departments regarding the events that occurred during the recent Clydesdale Bank Premier League match between Dundee United and Celtic. I can also confirm that Steven Craven has tendered his resignation and will no longer officiate in the professional game."
Regan added: "I hope to conclude the investigation within the next 48 hours and will make further comment at that time."
With the game on Tayside level at 1-1, McDonald awarded Celtic a penalty then controversially changed his mind after consulting with Craven. Despite scoring a last-gasp winner through Gary Hooper, Celtic boss Neil Lennon was furious at the referee's U-turn and the fall-out continued all last week.
Dallas made a statement applauding McDonald for arriving at the "right decision in the end" but Craven was reportedly angry that it was not made clear that, contrary to perception, he did not instigate the discussion with McDonald.
It was reported that since the fixture, Craven has been the victim of threatening remarks with his teenage sons suffering verbal abuse.
The official's resignation was anticipated and it was confirmed on Monday.