With its record $214.8 billion budget, California has become the first state in the U.S. to expand government-subsidized health care for some adult immigrants living in the country illegally, the latest example of the state forging its own path on policies related to immigration.

Democratic Gov. Gavin Newsom—whose party holds sizable majorities in the state legislature following the 2018 midterm elections—has signaled he would sign the state’s new spending plan, which amounts to a 7% increase over 2019’s budget. That funding...