Practicing medicine without a license or beyond the scope of your license (e.g. engaging in medical activities that your license does not authorize you to engage in, even though you have license to engage in some medical activities) is a criminal offense (e.g. Unlawful Conduct Of Practicing Medicine Without a License, Utah Code Ann. §58-1-501(1)(a) and 58-67-501) in most jurisdictions and is not protected by the privileges afforded to people practicing medically within the scope of their license (e.g. an exemption from laws criminalizing contact with intimate parts when done for medical purposes). Of course, fraudulently claiming to have licensure is also a crime over and above practicing without a license.
Administering non-FDA approved medicines to cancer patients, for example, is a federal crime, even if this is done with full disclosure and good intentions. The approval process is described here by the Food and Drug Administration under the Federal Food, Drug and Cosmetic Act. (Similarly, a conviction was obtained in another case for distributing a hormonal weight-loss treatment without a license.)
For example, in 2006, a naturopath in Wheat Ridge, Colorado was convicted of "theft, perjury, criminally negligent homicide, illegal practice of medicine and third-degree assault" for providing alternative holistic treatment to someone resulting in their death.
Criminal negligence generally involves conduct sometimes also called "gross negligence" that is not just careless but is almost reckless given the serious potential harm that could foreseeable result (and in all cases that are prosecuted, actually did result) from the course of conduct taken.
For example, while ordinary medical mistakes by a medical doctor such as confusing two drugs with similar names or putting the decimal point in a prescription dosage, causing harm to a patient, would not ordinarily result in criminal liability, coming into an operating room while too drunk to drive and without reviewing which limb of a patient needs to be amputated despite a clear indication in marker on the leg of a patient showing that fact, might constitute criminal negligence on the part of a medical doctor. Here, if the need for and possible benefits of conventional treatment for breast cancer with particularly clear, prescribing alternative diet based treatment and mental exercises while discouraging conventional medical treatment, might very well constitute criminal negligence on the part of the holistic practitioner.
Similarly, a naturopath was criminally charged in Australia with "reckless grievous bodily harm and failure to provide for a child causing danger of death" for urging parents to discontinue medical treatments for a child in favor of a raw food diet, causing serious harm to that child almost causing the child's death. The naturopath admitted that she endangered the child with her medical advice and was ultimately convicted in that case.
Liability would be fact specific. Does the person have a license of some kind? Are they within the scope of their license? Do they falsely convey the impression that they are licensed medical practitioners? Did their actions constitute the practice of medicine?
But, often, in the fact pattern you describe, particularly if it is not "faith healing" protected by the freedom of religion, this would be a crime.
Of course, this doesn't mean that a self-help remedy of murdering the holistic practitioner after the fact is legal.
Civil liability for professional negligence and failing to meet the applicable standard of care for a person engaging in the kind of treatment conduct described is also possible. In other words, one can sue a naturopath of money damages for malpractice, just as one could sue a doctor for malpractice. As noted here:
Alternative medicine providers can, of course, be guilty of
malpractice if they perform their interventions below the commonly
accepted standards of their own communities. They may also have
liabilities for injuries caused by discouraging patients from seeking
conventional care and, in some jurisdictions, for not recognizing when
a patient’s condition is beyond the scope of their form of treatment
and subsequently referring the patient for treatment by a medical
The fact scenario in the Slate article would appear to implicate grounds for civil as well as potentially criminal liability.