May received a one-match ban from the Demons for drinking while in the AFL's concussion protocols following an altercation with Melksham in Prahran on Sunday night, after the pair went for a dinner with teammates.
The star full-back lamented Sunday's "bad error of judgement" and said he'd have to prove himself to his teammates again.
Currently one of the league's best defenders, premiership winner May appeared to have left difficult moments in his career behind him.
In May 2019, May avoided a club sanction and apologised to his teammates for drinking while injured, amid a debut season at Melbourne that was ruined by injuries after turning up to pre-season unfit.
Early in his career at Gold Coast, May had his jaw broken by then-teammate Campbell Brown in an altercation at a Los Angeles nightclub in 2013.
"I know I've let a lot of people down, especially myself and the footy club, through uncharacteristic actions," May said in a video posted to Melbourne's website.
"I've done a lot of work to try and build respect and trust at the footy club, and I'm disappointed these actions don't reflect that.
"Just a bad mistake, a bad error of judgement, (which) unfortunately has consequences.
"Me not being out on the field is letting them (the fans) down, letting the team down. It'll be hard to watch the game knowing I'm responsible for not being out there.
"The hard work's going to come in the next six to 12 months where I prove my worth and how committed I am to the footy club."
May's indiscretion comes amid a tough period for the Demons, who have been putting spotfires out on all fronts.
Two weeks ago the reigning premiers were flying and appeared unbackable flag favourites.
They have since lost two games on the bounce while text messages between current president Kate Roffey and predecessor Glen Bartlett from 2020, featuring criticisms of the team, were on the back page of Tuesday's Herald Sun.
On-field, May's fellow defender Harrison Petty (shoulder) will have to prove his fitness for Monday and swingman Tom McDonald is sidelined until late in the season with a foot injury, while Jake Lever has struggled without May next to him in defence.
Melbourne have desperately missed star defender May, who was concussed early in the loss to Fremantle and unavailable in the defeat Sydney, but said they needed to maintain their standards.
"Over the past three years, Steven has developed into one of our most important on-field leaders, and while the impact of not having him available for selection is significant, what is more significant to us is our culture and the commitment we have to live by our values," Melbourne general manager of football performance Alan Richardson said in a statement.
Both May and Melksham, who avoided suspension but has been a fringe player this season, will also complete community service.
