Brown and his assistant Archie Knox resigned from their posts to join the Dons last week.
The pair were working without contracts but the Lanarkshire club felt Scottish Premier League rules had been breached.
However, they have now informed the SPL that their grievance has ended following "constructive discussions" with the Pittodrie club.
"During a difficult time for Scottish football and when clubs are about to enter discussions on a major league restructuring, we are of the view that it is not in anyone's best interests for two Scottish Premier League clubs to be involved in an ongoing dispute," read a statement from Fir Park.
"Following constructive discussions with Aberdeen, we have reached an agreement to draw a line under the matter and have formally withdrawn our complaint to the SPL in relation to their employment of Craig Brown and Archie Knox.
"We have always maintained a good working relationship with Aberdeen and look forward to that continuing in future.
"We strongly believe this action is in the best interests of the club and will make no further comment on the matter. We will now focus all our resources on employing a new management team to take the club forward."
Dons chairman Stewart Milne had earlier refuted claims his club acted inappropriately.
"At no point was the integrity of Craig or Archie questioned by either club and we believe that we acted accordingly throughout.