Milan have endured a nightmare season so far, and sit a lowly 11th in Serie A following a 4-2 home defeat to Parma on Sunday.
Fan protests preceded the defeat - Milan's fourth in a row - in which fans reportedly directed their anger at Galliani and striker Mario Balotelli.
And Maldini has also moved to blame Galliani for the club's dismal performance this term.
Maldini spent his entire career with Milan, winning seven Serie A titles and five UEFA Champions League crowns, but feels that Galliani's "failure to understand the players" has been a key factor in the club's decline.
Galliani was convinced to stay by club president Silvio Berlusconi in November after tendering his resignation, and is now eager to guide the 18-time Italian champions through this difficult spell.
"I'm not quitting," he told RAI. "I've been with Berlusconi for 35 years and I'll be by his side for as long as he wants.
"I've decided to remain silent, although I'm talking to the president every day."
Milan travel to Lazio on Sunday hoping to earn only their 10th win of the campaign.
