The state included a coronavirus vaccine among the required vaccinations in a regulation issued in August, setting a deadline of Oct. 29. Several health care workers sued, saying the requirement violated their constitutional right to the free exercise of religion.

Judge Jon D. Levy of the Federal District Court in Maine ruled against the plaintiffs.

“Both the serious risk of illness and death associated with the spread of the Covid-19 virus and the efforts by state and local governments to reduce that risk have burdened most aspects of modern life,” he wrote.

The plaintiffs’ “refusal to be vaccinated based on their religious beliefs has resulted or will result in real hardships as it relates to their jobs,” Judge Levy wrote. “They have not, however, been prevented from staying true to their professed religious beliefs which, they claim, compel them to refuse to be vaccinated against Covid-19.”

A unanimous three-judge panel of the U.S. Court of Appeals for the First Circuit, in Boston, affirmed Judge Levy’s ruling.

Judge Sandra L. Lynch, writing for the panel, said the regulation did not single out religion for disfavored treatment. In a 1990 decision, the Supreme Court ruled that neutral laws of general applicability that incidentally impose burdens on religion generally do not run afoul of the First Amendment’s protection of religious liberty. That decision, Employment Division v. Smith, has been the subject of harsh criticism by the more conservative members of the Supreme Court.

The plaintiffs in the case from Maine said the state was an outlier in refusing to grant religious exemptions.

“Almost every other state,” they told the justices, “has found a way to protect against the same virus without trampling religious liberty — including states that have smaller populations and much greater territory than Maine. If Vermont, New Hampshire, Alaska, the Dakotas, Montana, Wyoming, California and the District of Columbia can all find ways to both protect against Covid-19 and respect individual liberty, Maine can too.”