The apology came as Scottish Football League clubs discuss a proposal to put Murray's club in the Irn-Bru First Division if Scottish Premier League clubs reject the top-flight application on Wednesday.
Murray said in a statement on www.rangersfc.co.uk: "On behalf of the new board of directors, I apologise unreservedly to all for the distress, disruption and difficulty inflicted on the football community, caused entirely by the actions of people who failed miserably to act responsibly during their stewardship of Rangers."
Rangers were consigned to liquidation last month amid debts that could reach more than £100million, much of which is owed to the taxpayer.
Murray, whose company bought the assets and business of Rangers for £5.5million, said: "The last few months have been extremely difficult for everyone at Rangers - staff, players and, most importantly, the supporters of the club who have been magnificent in the face of adversity.
"That said, all of us within the Rangers family are acutely aware that events at our club, brought about by people who are no longer here, have triggered a crisis in Scottish football. Ally McCoist, the staff and players have nothing to apologise for but the club needs to make an apology. It is only right that someone at Rangers expresses our sorrow and regret."
Clubs such as Motherwell have warned that they could face an insolvency event because of the collapse of Rangers, even if they accept the newco club into the SPL.
Motherwell, who are due to confirm the result of their fans' vote, said the loss of Rangers in the SPL next season could cost them up to £900,000 but voting the newco in and alienating loyal fans would also lead to long-term financial damage.
Murray said: "All of Scottish football is now having to live with the consequences of that behaviour and every true Rangers fan will consider it a matter of deep regret that events at our club in the past have been the root cause of an extremely difficult time for Scottish football.
"The people who are now running Rangers Football Club want the club to move forward but we cannot do so until we have recognised how damaging to football has been the crisis at our club. We want to rebuild Rangers but we also want to ensure the club plays a role that is beneficial to Scottish football, the Scottish economy and Scottish life in general."