The former Gers manager stepped down from his non-executive role on Monday.
This follows the news that former chief executive Charles Green had been appointed in an advisory capacity last Friday.
In a statement Smith said: "It is with the greatest of regret that I have to announce to Rangers supporters that I have decided to step down as chairman of Rangers Football Club.
"It is, I am convinced the correct thing to do under the circumstances.
"Unfortunately, recent events have led me to the conclusion that the only option left open is for me to submit my resignation and to leave the board of directors with immediate effect.
"Working with a board which could rarely find consensus and agreement has created a highly-dysfunctional environment which has not been good for the club.
"On top of that, there has been recent further disruption which again has rendered the board less effective and efficient than it ought to have been.
"It is clear that boardroom change is required and I would urge Rangers fans to get fully behind the resolutions which last week were presented to the board and shareholders.
"Stepping away from this club is heart breaking but I will always be grateful for the support Rangers fans have given me down through the years."
Regarding, Craig Mather, the club's current chief executive, Smith went on: "As I have said before, he is doing a good job.
"It is unfortunate that this has happened but when change does come I hope he'll be allowed to continue the job he has started.
"As for Alistair McCoist, it is important that we all appreciate that no other Rangers' manager has had to work with poorer or more trying circumstances.
"It is my fervent hope that when the dust settles and Rangers have a clearer path and future, he is given the chance to manage under conditions similar to those which were afforded his predecessors.
"Last but most certainly not least, I must turn to Rangers supporters. They have suffered agonies and tortures over the last two years but they can be proud of their contribution.
"Indeed, they are a reminder to those in the boardroom that directors have the massive and ultimate responsibility of serving, protecting and nourishing this club for the benefit of the fans."
Mather, who replaced Green in April, added in a statement: "I am saddened that Walter felt he had to leave the board because his advice and common sense approach over the past few months have helped enormously.
"I am certain the last thing he would want to do is damage the club more than is necessary. After all, he returned to Ibrox to help rebuild the club, not to disrupt that process.
"However, I have to accept his decision and respect it. But even though he has stepped down he has made it clear that if I feel the need I can call him at any time."