The Brazil international was injured during the second half of Saturday's 2-2 draw at Torino - his first competitive game since he moved back to Italy after a four-year spell at Real Madrid.
Kaka sustained an adductor muscle injury in his left thigh and will undergo further tests in 10 days' time to confirm the exact recovery period.
"Yesterday was a tough day for me as I found out about my injury," he told acmilan.com.
"I certainly didn't expect this type of injury at this time. It has been hard and difficult.
"I spoke to the club a lot yesterday, to (vice-president Adriano) Galliani and to the doctors to decide on a few things.
"I have decided that I don't want anything from Milan except their (moral) support and affection.
"Until I am fit again and ready to play, I have decided not to receive my wages.
"I have already started the recovery process.
"The affection I have felt in recent days gives me the added motivation to get fit as soon as possible."
The 31-year-old Kaka rejoined the Italian giants earlier this month on a free transfer from Real Madrid and reportedly accepted a 45 per cent salary reduction.
He left the Bernabeu after four years - having joined from Milan for a reported £56million in 2009 - and received a hero's welcome from the Rossoneri supporters.
Kaka's injury is a further blow to Milan head coach Massimiliano Allegri, whose side open their Champions League campaign against Celtic on Wednesday.
The Rossoneri also lost captain Riccardo Montolivo (thigh) and midfielder Andrea Poli (leg) at the weekend while forward Stephan El Shaarawy is doubtful for Wednesday's game with a thigh injury.
Allegri is also without defenders Ignazio Abate, Mattia De Sciglio, Matias Silvestre and Daniele Bonera as well as striker Giampaolo Pazzini due to injury.