The Brazil international sustained an adductor muscle injury in his left thigh in 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, who will undergo further tests in 10 days' time to confirm the exact recovery period, confirmed on Monday his intention not to receive his wages until he returns to play.
"Kaka's decision came about during a phone call that I received from him where he was crying," Galliani told www.acmilan.com.
"He told me he wanted to feel the affection of Milan and to repay the fans for their support.
"He said he did not want a single euro until he returns to play.
"This act is evidence of the extraordinary rapport that Kaka has with the fans."
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.
"A few days ago Kaka was telling me of his joy at returning to Milan," Galliani said. "He started his second spell at Milan in the best possible way.
"Today's news is not Kaka's injury but rather his commitment to the club.
"Only 15 days ago he had reduced his salary by a lot and today he wanted to give another sign of commitment and that is fantastic."