Biden is requiring the vast majority of federal workers to get vaccinated or face disciplinary measures.

Slightly more than half of Americans, 53 percent, are now fully vaccinated. While the number of people seeking shots ticked up considerably in August, as Delta caused virus cases to soar, the vaccination rate has yet to help the nation achieve “herd immunity” — the tipping point that occurs when widespread vaccination, coupled with natural immunity, slows the spread of a virus. If it continues to spread, officials fear that it will mutate into another, even more dangerous variant that evades vaccines.

“When you have 75 to 80 million people who are eligible to be vaccinated, who don’t get vaccinated, you’re going to have a dynamic of continual smoldering spread of the infection,” Mr. Biden’s top medical adviser for the coronavirus pandemic, Dr. Anthony S. Fauci, warned in an interview, adding, “It’s very frustrating, because we have the wherewithal within our power to be able to actually suppress it.”

The mandate for federal workers is an especially assertive move by the president. Jen Psaki, the White House press secretary, told reporters on Thursday that, aside from some religious and disability exemptions, the vast majority of federal workers would be subject to a 75-day grace period for receiving a vaccine.

If workers decline to receive shots in that time frame, Ms. Psaki said, they will “go through the standard H.R. process,” which she said would include progressive disciplinary action. At least one major labor union challenged the mandate even before Mr. Biden delivered his speech.

Cathie McQuiston, a deputy general counsel for the American Federation of Government Employees, a union representing some 700,000 federal workers, said in an interview that her organization would be working with agencies to “not skip over procedures and make sure employees have due process” if they were disciplined.

It is unclear how long it will take for the new requirements to go into effect, and some will almost certainly give rise to lawsuits. However, Lawrence O. Gostin, a professor of global health law at Georgetown University, said Mr. Biden has broad latitude to compel vaccination, even for workers in private businesses.

“The federal government has ample power to regulate health and safety in the workplace,” Mr. Gostin said. “Employers have a legal obligation to comply with evidence-based federal health safety standards. Clearly, Covid-19 vaccinations are vital to ensure worker safety and could be required.”