They claim Facebook endangers the health of people around the world and are calling for a better policy for the future naming of public facilities.
The resolution has no basis in law and is largely symbolic. The full board is expected to vote on the resolution on 15 December.
“San Francisco’s only public hospital should not bear the name of a person responsible for endangering public health in our country and around the world — and yet it does,” said Gordon Mar, the resolution's lead signatory.
“These are policy choices, and they have a body count.”
Mr Zuckerberg and Ms Chan donated $75 million to the San Francisco General Hospital, where Ms Chan worked as a paediatrician, in 2015.
As part of the gift process, the medical centre was officially renamed the Priscilla Chan and Mark Zuckerberg San Francisco General Hospital and Trauma Center.
But since then, Facebook has come under increasing criticism for the way it conducts its business, perhaps most notably over the Cambridge Analytica scandal.
“We’re of course thankful for the gift and we’re thankful for any gift to our most important institutions during this time,” said Matt Haney, another supervisor supporting the condemnation.
“But that doesn’t mean that we should for forever essentially have given away advertising rights on this most essential public institution.”
Kim Meredith, the head of the hospital’s foundation, criticised the resolution, saying she was "proud" the hospital was named after Mr Zuckerberg and Ms Chan.
She said: “This resolution of condemnation on the naming of [the hospital] has the potential of unintended consequences and a chilling effect on past, present, and future gifts to the city.
“We will need philanthropies to continue to tackle the challenges of Covid-19, health equity, and recovery in future years.”
The Independent has contacted the Chan Zuckerberg Initiative for comment.