The proposal was introduced Thursday by GOP lawmakers in the state Legislature. Universities that allow violence to happen would also be penalized if the proposal were to become law.
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Supporters of the legislation say school administrators should also be fired for what they allow to happen on their campuses.
The proposal faces long odds in the Democratic-controlled Legislature.
The bill was introduced on the same day that officers surged against a crowd of demonstrators at UCLA, ultimately taking at least 200 protesters into custody after hundreds defied orders to leave, some forming human chains as police fired flash-bangs to break up the crowds. Police tore apart a fortified encampment's barricade of plywood, pallets, metal fences and dumpsters, then pulled down canopies and tents.
Like at UCLA, tent encampments of protesters calling on universities to stop doing business with Israel or companies they say support the war in Gaza have spread across other campuses nationwide in a student movement unlike any other this century. Iranian state television carried live images of the police action at UCLA, as did Qatar's pan-Arab Al Jazeera satellite network. Live images of Los Angeles also played across Israeli television networks.
Israel has branded the protests antisemitic, while Israel's critics say it uses those allegations to silence opposition. Although some protesters have been caught on camera making antisemitic remarks or violent threats, protest organizers - some of whom are Jewish - call it a peaceful movement to defend Palestinian rights and protest the war.
Long-term consequences? What students need to know about protesting on campus
President Joe Biden on Thursday defended the right of students to peaceful protest but decried the disorder of recent days.
UCLA chancellor reacts to confrontations on campus
The confrontations at UCLA also played out over several days this week. UCLA Chancellor Gene Block told alumni on a call Thursday afternoon that the trouble started after a permitted pro-Israel rally was held on campus Sunday and fights broke out and "live mice" were tossed into the pro-Palestinian encampment later that day.
In the following days, administrators tried to find a peaceful solution with members of the encampment and expected things to remain stable, Block said.
That changed late Tuesday, he said, when counterdemonstrators attacked the pro-Palestinian encampment. Campus administrators and police did not intervene or call for backup for hours. No one was arrested that night, but at least 15 protesters were injured. The delayed response drew criticism from political leaders, including Gov. Gavin Newsom, and officials pledged an independent review.
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"We certainly weren't thinking that we'd end up with a large number of violent people, that hadn't happened before," Block said on the call.
By Wednesday, the encampment had become "much more of a bunker" and there was no other solution but to have police dismantle it, he said.
The hourslong standoff went into Thursday morning as officers warned over loudspeakers that there would be arrests if the crowd - at the time more than 1,000 strong inside the encampment as well as outside of it - did not disperse. Hundreds left voluntarily, while another 200-plus remained and were ultimately taken into custody.
Pro-Palestinian protests continue at SoCal college campuses
Meanwhile, protest encampments at other schools across the U.S. have been cleared by police - resulting in more arrests - or closed up voluntarily. But University of Minnesota officials reached an agreement with protesters not to disrupt commencements, and similar compromises have been made at Northwestern University in suburban Chicago, Rutgers University in New Jersey and Brown University in Rhode Island.
A UCLA student's perspective
Ariel Dardashti, a graduating UCLA senior studying global studies and sociology, said no student should feel unsafe at school.
"It should not get to the point where students are being arrested," Dardashti said on campus Thursday.
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Dardashti, who is Jewish, said he can relate to the trauma suffered by Palestinians.
"When my dad was fleeing Iran, he prayed that his children wouldn't have to face antisemitism," Dardashti said. "We're afraid of having to flee again in the same way our parents did."
The Associated Press contributed to this report.