A meeting of the Saints board on Tuesday night decided that its duty of care to Milne and his family dictated that the forward should not take the field.
Milne, 33, faces four charges of rape over an alleged incident in 2004.
A statement by St Kilda read, "The board acknowledges this is a most serious matter, one that is distressing to all the parties involved.
"It also recognises that the extended time-frame of the matter and the complexities of the issues involved, make it without precedent.
"In considering the matter the board has been mindful of the rights and obligations of everyone impacted.
"Based on this the board has resolved that in the best interests of all parties concerned, and the duty of care the club has to Stephen and his family, Stephen will remain a listed player but will take a leave of absence from playing for an indefinite period.
"The St Kilda Football Club respects that this matter is now before the courts. As a consequence we won't be making any further comment at this time."
Earlier Milne trained with teammates, joining the group on the track just after 10.30am.
He had avoided waiting media when he arrived at the club.
Milne had been "rocked" by the charges laid on Tuesday but wanted to play against Melbourne on Saturday, Saints captain Nick Riewoldt told Melbourne radio on Wednesday morning.
Riewoldt backed the veteran's plan to appear against the Demons, saying that Milne had been affected by the charges laid, but believed he could contribute against Melbourne at the MCG.
"I think he should [play this weekend]," Riewoldt said.
"'Milney' is the absolute heart and soul of the football club and everyone knows how passionate he is and what the football club in particular means to him.
"I think given how unique these circumstances are … it's really up to him.
"If he feels like he can contribute and he can put it to the side and go out for two hours, which he's stated unequivocally that he's going to be able to do and he wants to do, then I think that's where it should stay."
Midfielder Nick Dal Santo said before the club's statement was issued that he also hoped Milne would take the field.
"He's a quality person and we support him fully, and it'd be great to have him out there."
Milne and teammate Leigh Montagna were investigated for an alleged incident involving in a 19-year-old girl in 2004, but have consistently denied any wrongdoing.
Riewoldt said he understood why many people would view Milne's desire to play this weekend in a negative light.
"Certainly I understand because this a difficult time for everyone," he said.
"As a close mate of Milney's, you see what he goes through, and there are other parties involved that it's going to be as equally difficult for.
"I understand that some people will disagree with that line of thinking and that's the nature of people's opinions, I guess."
He said Milne was "incredibly resilient" but there was no doubt he had been affected by the proceedings.
"Clearly it's rocked him but … this has been hanging over his head in one way, shape or form for nine years," he said.
"I think to do what he's done over that period of time on the field is a credit to him and the support he's had from his family.
"He's a very resilient guy, but no doubt he's thinking a lot about it and it's rocked him."
The Saints' board is believed to have met on Tuesday night although it has been reported CEO Michael Nettlefold is overseas.
The clash with Melbourne is Riewoldt and Dal Santo's 250th game and Sean Dempster's 150th.
Jennifer Phelan is a reporter for AFL Media. Follow her on Twitter @AFL_JenPhelan.