Some public health experts worry that the administration is not being aggressive enough in waging what the president calls a “wartime effort” to vaccinate the country.WASHINGTON — Faced with a steep decline in vaccination rates, President Biden said on Tuesday that his administration would send people door to door, set up clinics at workplaces and urge employers to offer paid time off as part of a renewed push to reach tens of millions of unvaccinated Americans.But top health experts say that it is simply not enough, and that the president needs to take the potentially unpopular step of encouraging states, employers and colleges and universities to require vaccinations to slow the spread of the coronavirus.Instead, in a speech on Tuesday, Mr. Biden doubled down on coaxing people to get vaccinated, a voluntary approach that appears to have hit its limit for a large number of Americans who say they have no intention of taking the shot.“Please get vaccinated now. It works. It’s free,” Mr. Biden said in brief remarks at the White House. “It’s never been easier, and it’s never been more important. Do it now for yourself and the people you care about, for your neighborhood, for your country. It sounds corny, but it’s a patriotic thing to do.”Case numbers have gone up in parts of the country where vaccination rates remain low, fueled by the highly contagious Delta variant. That has some public health officials worried that the administration is not being aggressive enough in waging what the president calls a “wartime effort” to ensure that the population of the United States is protected.“I’m trying to restrain myself, but I’ve kind of had it,” said Kathleen Sebelius, who was the health secretary for five years under President Barack Obama. Schools and businesses should be encouraged to require the vaccine, she said.“You know, we’re going to tiptoe around mandates,” she said. “It’s like, come on. I’m kind of over that. I want to make sure that people I deal with don’t have it so I don’t transmit it to my granddaughter.”But Mr. Biden’s options to be more aggressive are limited.As president, he can mandate that members of the military get the vaccine — a step that his administration has declined to take, in part because the drugs are still considered experimental under the emergency authorizations that the Food and Drug Administration granted last year.The Biden administration considered and rejected calls to require a federal vaccine passport, a move that some experts said would help contain the spread of the virus by allowing people to prove that they have been inoculated. And the administration last month issued guidance to federal agencies saying they should not require employees to be vaccinated.For the most part, the power lies in the hands of states, employers or private institutions.Dr. Ezekiel J. Emanuel, a professor of bioethics and health policy at the University of Pennsylvania, said the United States was unlikely to make significant strides in its vaccination campaign without mandates.“I like to say a mandate is legal, ethical and efficacious,” he said. “Ultimately, workplaces are probably going to have to.”In his speech, Mr. Biden said his administration was not giving up on persuading people that vaccination was in their best interests, and in the interest of the country. But he made no mention of the need for states, private companies, schools and other institutions to begin requiring people who were reluctant to get vaccinated.Jen Psaki, the White House press secretary, acknowledged in comments to reporters on Tuesday that some companies, schools and other institutions were beginning to require vaccines. But she said the administration had no intention of encouraging them to do so.“We’re going to leave it up to them to make these decisions,” Ms. Psaki said.But others say the administration could be more aggressive.Lawrence O. Gostin, a professor of global health law at Georgetown University, said that even though the federal government’s authority to enact mandates was limited, the Biden administration still had considerable power to recommend them. It can provide more funding for proof-of-vaccination systems and create incentives for colleges, universities and organizations to require that a vaccine be offered, he said.“Vaccine mandates have been very successful in the United States and globally, even in politically difficult situations, because they make becoming vaccinated the default,” Mr. Gostin said. “We have to make being unvaccinated the hard choice, not the easy one.”The debate comes as Mr. Biden and the United States face a precarious moment days after narrowly missing his goal of having 70 percent of adults at least partly vaccinated by July 4. By the end of the week, nearly 160 million Americans — not quite half the population — will be fully vaccinated. But vaccination rates have plunged from where they were in the spring, and some parts of the South and the Midwest continue to struggle to inoculate their populations.Laura Alfredo received her first dose of the Pfizer vaccine in Newark last month. Vaccination rates have plunged from their peak in the spring.Bryan Anselm for The New York TimesAlabama has vaccinated only about 50 percent of its adult population; Mississippi has delivered shots to only 46 percent of its adults. At their current rates, it would take months for both states to reach Mr. Biden’s July 4 goal. Louisiana and the Virgin Islands have each vaccinated fewer than half of their populations.Numbers were down across the United States: As of Tuesday, providers were administering about 0.87 million doses per day on average, a 74 percent decrease from the peak of 3.38 million doses reported on April 13.That reality prompted Mr. Biden to announce what he called a renewed push to increase the number of vaccinated Americans. All of the steps he outlined in his speech are avenues his government has already pursued: door-to-door outreach to get Americans vaccinated; a push to provide vaccines to primary care doctors; bolstering efforts to get vaccines to pediatricians and other providers who serve younger people so that adolescents ages 12 to 18 can get their shots; expanded mobile clinic efforts and workplace changes.“Please, please, get vaccinated,” Mr. Biden said on Tuesday. “It makes a big difference.”The question of whether pleading is enough is at the heart of what the government and private businesses do next in the effort to slow the spread of the virus.States have broad authority to require vaccinations, including among health care workers, though they generally have not mandated vaccines for adults or for Covid shots.All 50 states require certain vaccines for children who attend school, but those mandates apply only to vaccines that have been fully approved by the Food and Drug Administration, a status the coronavirus shots have not yet reached. Any state mandates for Covid vaccines would almost certainly allow students to opt out for medical, religious and sometimes philosophical reasons, as they do for other childhood shots.Many state mandates to vaccinate schoolchildren were first imposed in the 1960s, at the urging of the Centers for Disease Control and Prevention after measles outbreaks.The idea of requiring vaccinations as a condition of employment or status as a student can be a politically volatile idea, which could backfire on Mr. Biden if he embraces it too aggressively, Dr. Emanuel said.The simple act of mask-wearing, he said, had become highly politicized and did not involve a new vaccine.“You just have to be realistic about what’s possible,” Dr. Emanuel said. “If the federal government does it, there’s going to be a ton of backlash. It’s going to be a political event. If private industry does it, it’s like, ‘OK, that’s private industry, that’s what we’re founded on.’”The Health Innovation Alliance, a coalition that supports the use of data and technology to improve outcomes, asked the White House this spring to set standards for digital “vaccine passports” — applications that can verify if a person is vaccinated — but that push gained little traction.“The government is not now nor will we be supporting a system that requires Americans to carry a credential,” Ms. Psaki said at the time. “There will be no federal vaccinations database and no federal mandate requiring everyone to obtain a single vaccination credential.”And when the White House hosted 1,000 military personnel and essential workers for a Fourth of July bash on the South Lawn on Sunday, proof of vaccination was not required, to the dismay of public health experts.“There is so much toxic politics around Covid that it’s constraining sensible action,” Dr. Tom Frieden, a former C.D.C. director, said in an interview last week. “Obviously it makes sense to require proof of vaccination in various settings, but that has become a political lightning rod.”Studies have shown that many Americans are anxious about taking a new vaccine under so-called emergency use authorization, seeing the review system as rushed. They are waiting for the Food and Drug Administration to fully approve a vaccine before taking it. A recent Kaiser Family Foundation survey found that three in 10 unvaccinated adults said they would be more likely to accept a vaccine if one were to be approved.But approvals are considered unlikely until at least September, according to people familiar with the F.D.A. review process. Regulators are already working to conduct a review that typically takes at least 10 months in half the time or less.Dr. Steven Joffe, a bioethicist at the University of Pennsylvania, said a full approval would have significant influence on how the public perceived the safety of vaccines and would most likely increase inoculation rates. In addition to making vaccinations more convenient, he said, Mr. Biden should use the bully pulpit to support more employer vaccine requirements.“The federal government encouraging that, and norm-setting, would be a very good thing,” Dr. Joffe said. “The risk is when they start to push things too hard, it gets into politically charged territory, whatever the legality of it.”Sheryl Gay Stolberg