Assm. Paul Fong, D-Cupertino, authored the bill. He was joined Monday by environmentalists and several San Francisco mayoral candidates in support of the legislation.
The ban's supporters dismiss accusations the measure unfairly attacks Chinese culture, where the soup is considered a delicacy.
"We will not miss it for its taste, because it's tasteless, we will not miss it for its nutritional value because there is none and we will not miss it for its cultural significance because there isn't any; what we will miss will be sharks," Reef Check Foundation spokesperson Sue Chen said.
It's estimated 73 million sharks are killed each year for their fins.
The governor's staff says they've been hearing from both sides on the issue. Brown has until Oct. 9 to sign it into law or veto it.
Last July, Hawaii became the first state in the country to ban shark fin soup.