LOS ANGELES — A bill that would require nearly all children in California to be vaccinated by eliminating “personal belief” exemptions advanced through the State Legislature on Wednesday, though it still has several hurdles to clear. If approved, California would become one of only three states that require all parents to vaccinate their children as a condition of going to school, unless there is a medical reason not to do so.

Under the bill, introduced after a measles outbreak that began at Disneyland, parents who refuse vaccines for philosophical or religious reasons would have to educate their children at home. The legislation prompted a roiling debate in Sacramento, and last week hundreds of people protested at the Capitol, arguing that it infringed on their rights and that it would unfairly shut their children out of schools.

Last Wednesday, the legislation stalled in the Senate Education Committee as lawmakers said they were concerned that too many students would be forced into home schooling. This Wednesday, however, the bill passed that committee after its authors tweaked it, adding amendments that would expand the definition of home schooling to allow multiple families to join together to teach their children or participate in independent study programs run by public school systems.

“We think we’ve struck a fair balance here that provides more options for parents who don’t want to vaccinate their children,” said Senator Ben Allen, a Democrat from Santa Monica and a co-author of the bill.