May had surgery on Saturday to insert two plates into his jaw after an altercation with Brown in Los Angeles on Thursday.
Suns chairman John Witheriff penned an open letter to supporters on Sunday, which said that Brown had accepted his suspension from the club.
"The full board of the Gold Coast Suns met on Thursday afternoon where the incident was discussed at length. Highlighting the seriousness with which the club is treating the issue, the board agreed to delegate its authority to make a decision on the matter to two board members, myself, and Mr Bob East," Witheriff wrote.
"Since the meeting of the board, chief executive Travis Auld and general manager of football operations Marcus Ashcroft have been conducting a detailed investigation of the incident.
"As you would expect, their brief has been to gather all the relevant facts, ensuring that any ruling the club makes will be made with accurate and substantiated information.
"This is being done in a timely manner, but I must stress that the board is committed to a fair and appropriate outcome. The integrity of the investigation is my dominant concern.
"Following the board meeting, the club notified Campbell Brown that he would be suspended from the club until the completion of the investigation. Campbell understands the suspension and is cooperating fully with the on-going investigation."
The club expects May to resume full training in two weeks.
Gold Coast suspend Campbell Brown over Steven May incident
