Already in deep distress, the patient was rushed last week to a hospital in Northern California, severely ill and unable to breathe on her own.

Doctors at the University of California, Davis Medical Center, near Sacramento, provided the woman with critical care but also considered an unlikely diagnosis: infection with the coronavirus.

Hospital administrators said they immediately requested diagnostic testing from the Centers for Disease Control and Prevention, but the procedure was not carried out because the case did not qualify under strict federal criteria: She had not traveled to China and had not been in contact with anyone known to be infected.

The announcement on Wednesday that the woman was indeed infected left health officials in California searching for people she may have exposed to the virus and testing the medical workers who have treated her. The case has raised difficult questions about whom to test and whether the nation is prepared to keep the virus under control.

The California woman’s case may also offer the first indication that the virus has spread beyond Americans who traveled outside the country, or had contact with someone who had.

Even before the announcement on Wednesday, frustration had been mounting among health providers and medical experts that the agency was testing too few Americans, which may slow preparations for an outbreak and may obscure the scope of infections.

“I think the diagnostic issue is the single most important thing that keeps me up at night right now,” said Lauren Sauer, director of operations at the Johns Hopkins Office of Critical Event Preparedness and Response in Baltimore.

C.D.C. officials said on Thursday that they had been unaware that doctors in California made an urgent appeal for diagnostic testing of the woman. But by the end of the day, the agency had revised and broadened its testing criteria, adding to the number of Americans who qualify.

In California, health officials are tracing close contacts of the woman, who lives in Solano County but has not otherwise been identified. Health care workers who have treated her are being monitored for the infection, and some employees at the medical center have been told to stay home.

Officials are bracing for a larger outbreak in Northern California. “There’s almost assuredly going to be a significant number of people testing positive,” said Dr. Peter L. Beilenson, the director of Sacramento County’s Department of Health Services.

The case has heightened concerns about the nation’s ability to test large numbers of people. Only the C.D.C. performs the tests that confirm a novel coronavirus diagnosis, a process that often takes days.

The C.D.C. had distributed diagnostic testing kits to state health departments, but they turned out to be flawed. Replacement kits have not yet been distributed.

Ms. Sauer said Johns Hopkins had treated several patients who did not fit the testing criteria, and for whom it requested coronavirus testing. In all but one case, the hospital was able to persuade the C.D.C. to run a test, or eventually identify another cause for the patient’s illness.

The C.D.C. is “pretty much the only place we can access testing,” she added, and the agency has been unwilling to grapple with cases that do not fit its criteria.

“The idea that we would have to really fight to get that test done, when C.D.C. is saying they have capacity, is alarming.” Ms. Sauer said. “It is a challenge when the most important piece of information — does this person have this disease, yes or no — is not accessible, and there’s no timeline for improved accessibility.”

After the diagnosis in California, the C.D.C. has pledged to greatly expand the state’s ability to test patients for the coronavirus, officials said.

“Testing protocols have been a point of frustration for many of us,” Gov. Gavin Newsom said at a news conference on Thursday. California had just 200 testing kits left, he added.

The governor said Dr. Robert R. Redfield, the director of the C.D.C., promised that state physicians would have a much greater ability to test patients who were showing symptoms, a change the governor said “can’t happen soon enough.”

The C.D.C. has committed to sending a team to California to help track people and make sure they are contacted by health officials about their possible exposure, Mr. Newsom said.

“They are being interviewed — points of contact, family members and others,” he said.

Experts said they were perplexed by the C.D.C.’s inability to fix the test’s flaws.

“The obvious observation is that many countries are capable of testing rather widely,” said Dr. William Schaffner, an infectious disease specialist at Vanderbilt University Medical Center in Nashville. “Why can’t we?”

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The C.D.C. operates two laboratories that test for the coronavirus and can handle approximately 400 specimens per day. Agency officials say there is no testing backlog, but it is unclear whether the labs will be able to keep up with demand if the need — and eligibility — increases testing substantially.

Under the new federal criteria, people with respiratory symptoms who traveled to Iran, Italy, Japan and South Korea should be tested — not just those who traveled in China. So should severely ill patients with acute lower respiratory symptoms who are hospitalized and in whom other diagnoses have been ruled out.

A criticism of the new criteria, however, is that doctors will have to wait until someone is extremely ill to test for the virus if that person did not travel to the affected regions or have contact with a known case.

“If we could identify these people earlier who don’t specifically meet one of the two criteria, or some sort of broader travel criteria, we could get them tested,” Ms. Sauer said. “You have to wait until someone’s really sick to push that test now, even with this new criteria.

Kenneth E. Raske, president of the Greater New York Hospital Association, said he planned to appeal to Vice President Mike Pence — whom President Trump named to lead federal preparations — “to order the C.D.C. to develop a rapid point of care test” that hospitals could use to screen patients.

In the meantime, Mr. Raske said, the Wadsworth Center, New York’s public health reference laboratory, should be certified to do these tests.

If more community-acquired infections turn up, and the disease cannot be contained, the strategy will have to become one of mitigation, said Dr. Neil Fishman, associate chief medical officer for the University of Pennsylvania Health System.

“That’s a little difficult to do when you don’t have a readily available test, and when the turnaround time for the test can be days instead of hours,” he said.

Thomas Fuller contributed reporting from Solano County, Calif., Nicholas Bogel-Burroughs from New York and Michael D. Shear from Washington, D.C.