Representatives for Mr. Newsom, a Democrat, didn’t respond to a request for comment.

The truck protests, originally intended to last a few days, have swelled since they began Monday, bringing cargo movements at the port almost to a standstill.

Protesters have prevented most trucks from dropping off and picking up containers at cargo-handling facilities. One trucker said protesters threatened to smash his windows if he entered the port.

As the number of protesters grew on Tuesday and Wednesday, unionized dockworkers refused to cross picket lines on foot or by car, citing health and safety concerns. The International Longshore and Warehouse Union has said it supports the new California law because it protects worker rights and helps them join a union.

Ed DeNike,

president of SSA Containers, which handles about 70% of the cargo entering and leaving the port, said he hopes to resume some operations soon.

Once dockworkers do get to their cranes they won’t be able to do much work. Mr. DeNike said his container yards are so full the firm has only enough space to move about 2,000 of the 10,000 containers it usually loads and unloads from ships in a 24-hour period. “At some point, we may not be able to do anything,” he said.

Port officials said 15 container ships were waiting for berth space by the docks on Wednesday.

The Oakland port, a major import gateway and a hub for U.S. agricultural exports, has struggled to handle surges of container volumes during the Covid-19 pandemic. Before the pandemic it was unusual for ships to have to wait for a berth at the port.

Peter Friedmann, executive director of the Agriculture Transportation Coalition, a Washington-based association representing farming interests in shipping, said the protests are a blow to exporters of almonds, walnuts, rice and dairy products who were just beginning to make progress after two years of struggling to get products overseas. “The supply chain already is in crisis,” Mr. Friedmann said. “This is a huge disruption.”

The protests at the port gates mark the latest disruption to hit U.S. ports since the impact of the pandemic upended trade flows in 2020, triggering big backups at Southern California ports as companies rushed to restock inventories and the gateways struggled to handle surging container volumes. The backup at the ports of Los Angeles and Long Beach has receded but bottlenecks have hit ports including Oakland at times and have more recently grown at East Coast ports as importers have sought alternative gateways.

The California law, known as AB5, is part of a broader regulatory battle across the U.S. over independent-contractor arrangements at transportation companies such as

Uber Technologies Inc.

and Lyft Inc. The International Brotherhood of Teamsters, which aims to organize drivers in California, says trucking companies have misclassified drivers as independent contractors to deprive them of fair wages and benefits.

Many trucking companies employ their own drivers. But in California they also rely on an estimated 70,000 independent owner-operators carrying loads between the state’s ports and distribution centers. The law makes it harder for trucking companies to classify the drivers that work regularly for them as independent contractors.

Many drivers say the new law will force them to seek work as an employee-driver or they will have to pay more for insurance and permits to remain independent under the law’s guidelines.

Said Ahmedi, a 47-year-old trucker and protester from Elk Grove, Calif., said he likes the freedom of being self-employed, but it would be too complicated to continue as an owner-operator under the new law.

Mr. Ahmedi said he earned almost $70,000 last year as an independent driver and was still able to take two, monthlong trips to visit family and in-laws in Afghanistan and Azerbaijan.

“If I am a working driver I am stuck with the company,” he said.

California legislators passed the law in 2019 and it was due to take effect in 2020, but it was delayed because of a legal challenge by the California Trucking Association. A June 30 decision by the U.S. Supreme Court not to review the case allows California to begin implementing the law in the coming months. The trucking association’s legal challenge continues in lower courts.

Write to Paul Berger at Paul.Berger@wsj.com