The Constitution gives states inherent "police power" to protect public health and safety. It is a broad power; however, the 14th Amendment prevents states from infringing on "the privileges or immunities of citizens of the United States" without due process of law.
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"All persons born or naturalized in the United States, and subject to the jurisdiction thereof, are citizens of the United States and of the State wherein they reside. No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws."
Under their reserved powers, states can create laws to promote public safety - known as "police powers." However, the Fourteenth Amendment limits those laws by requiring that they not infringe on a person's constitutional rights without due process.
The Tenth Amendment gives states all powers not specifically given to the federal government, including the power to make laws relating to public health. But, the Fourteenth Amendment places a limit on that power to protect people's civil liberties.
No. Public health regulations cannot violate a person's constitutional rights. However, the public health powers granted to the states are broad. Governors can order quarantines during a public health emergency or direct people to stay in their homes, as long as there are exceptions for food and other necessities. They can also impose curfews in the name of public health. There is even Supreme Court precedent for vaccine mandates.
State and local governments have a constitutional duty to protect public safety. However, other parts of the Constitution, such as the 4th and 14th Amendments, limit how far those powers can go.