Under reforms called for by Mr Schwarzenegger's education minister Richard Riordan, principals would become general managers, taking over staffing and funding duties that are currently handled by education authorities.
"You can't run a business unless someone is on top of that business," Mr Riordan told California's schools chiefs earlier this month. "If principals see a problem, they can change it. They don't have to ask anyone."
Strengthening headships is part of Mr Riordan's blueprint to devolve decision-making in California's 8,000-school public education system. The state's $42 billion (pound;22bn) education budget would also be distributed more directly to schools, wresting control from local officials.
Mr Schwarzenegger will determine whether to endorse the plan next month, said aides, when it may also come before California lawmakers.
Complaints that heads frequently carry the can for school performance while enjoying only limited managerial power, have been blamed for the recent recruitment malaise.
"The biggest impediment to improving student achievement is not being able to control teaching staff," said Mark Leos, assistant principal at Fairfax high school in Los Angeles. But he called the empowerment proposals a "mixed bag".
"I like the idea of giving principals more authority, but I don't know how much that will pull them away from being educational leaders, which is their primary responsibility."
Other heads said that they already had enough on their plate and feared additional duties could bog them down in bureaucratic micro-management.
"It's a little overwhelming, a lot of us are not trained (in the proposed new tasks)," said Steve Van Zant, principal of Aviara Oaks middle school, near San Diego.
Meanwhile, recommendations that California's primary and middle school science teachers limit the time they devote to hands-on instruction have caused consternation among many educators.
New standards, framed by the California Curriculum Commission, decree that laboratory experiments and other practical exercises in textbooks prescribed to support science teaching "compose no more than 20 to 25 per cent" of lesson time.
Developments in America's most populous state are considered a barometer for broader education trends. Its textbook-purchasing power means this recommendation could have wider repercussions if other states follow suit.
The textbook stipulations are currently under review by California's board of education, which will determine whether to adopt them next month.
"It's contrary to everything I know about how students learn, and is fundamentally in conflict with how science is done, (which is through) exploration, uncovering and understanding," said Bernard Khoury, executive officer of the American Association of Physics Teachers.