At a news conference in Sacramento, the governor said he also had been a victim during his own hospital visits, although he did not specify when or where.
Each time he left an operating room for various surgeries, Schwarzenegger said he was told "there were kind of, you know, people going through the file that had white coats on. And so obviously they snuck into the hospital, they did not fit in with the hospital staff at all. Those things happen."
The governor said his administration would work with hospitals to improve patient protections.
"If they're celebrities, if they are business executives, or just ordinary citizens, everyone's file ought to be protected," he said.
The identity of the woman responsible for looking at Shriver's files was not released. She is the same employee who sneaked into actress Farrah Fawcett's medical records, officials told the Times on Sunday.
That worker was fired in May 2007 after UCLA learned of the widespread breaches, but patients were not notified, the hospital said.
In all, the woman improperly looked at 61 patients' medical records in 2006 and 2007, according to state and local medical officials. Fawcett and Shriver were among 32 celebrities, politicians and other well-known people, the newspaper said. Names of the other patients were not disclosed.
Besides being California's first lady, Shriver is a former NBC newswoman and a member of the Kennedy family.
The head of the UCLA Hospital System, Dr. David Feinberg, apologized for the breaches and said the woman behind them had been a "rogue" employee.
Fawcett is battling cancer. Her attorney, Kim Swartz, said last week that after an employee at the hospital accessed Fawcett's medical records, details about her treatment showed up in the National Enquirer.
But Feinberg told the Times the hospital reviewed the fired employee's e-mails and phone calls and found no evidence any confidential medical information was shared inappropriately.
The secretary of the California Health and Human Services Agency, Kim Belshe, said Sunday that her agency is "very concerned about what appears to be a pattern of repeated violations."
The state will be taking action against UCLA, she said.
UCLA did not let state officials know about the breaches last year. Kathleen Billingsley of the Center for Healthcare Quality said a state investigator on Friday came across a document with the names of those patients affected.
Feinberg said hospital officials initially concluded that alerting authorities and the patients involved was not required. They are reconsidering whether to notify the patients because of the recent disclosures, he said.
The news of the snooping into Fawcett's medical records became public Wednesday, a few weeks after the hospital announced that several employees were fired for peeking at pop star's Britney Spears' files.
Schwarzenegger said he called on his administration to act after the Spears case became public last month.