Wiggs was elected as a director on 30 March, but his resignation was confirmed on Wednesday.
The Supercars chairman was poised to become RA chairman before a fallout over his chief executive plans, according to a report from the Sydney Morning Herald.
In a statement, RA chairman Paul McLean said: "Peter has decided to step down from the board and I understand his reasons.
"He has undertaken some very important work and has made a valuable contribution to the organisation, in a very short time, and we are thankful for his contribution.
"The immediate priority of the board is to install a replacement for Peter, and an interim chief executive. I will provide an update on those matters at the appropriate time."