The Strange Case is fantasy about the limits of medical science, used as a lever into moral and mental exploration. At the time of publication, medicine (especially drugs) was such an experimental field that doctors were essentially testing these drugs on their patients—many of which (laudanum, cocaine, oxycontin) were highly addictive. Addiction was known, but straddled the line between disease and vice. This parallel between Dr Jekyll’s fictional transformative drug and real drug use is the biggest component of medicine as a moral element. The story admits Hyde as an immoral creature of vice, symbolizing the degrading effects that addiction has on a person’s character, but Jekyll’s identity is retained (until he finally loses control of the transformations)—which, as his more reasonable persona, the guilt, abdication of conscience, and other tendencies that addicts are prone to. Jekyll is not as starkly immoral as Hyde, but he is certainly irresponsible–which is where the line for him starts to blur between physician practitioner and patient (more accurately, test subject).
Lanyon represents the establishment criticism of Jekyll’s risky and short-sighted experimentation, which traces its history through to the modern day. Doctors using drugs is no longer frowned upon (although, the precise drugs that were prescribed at the time of publication are now known as unmedicinal), but there is a clear establishment, and it’s represented by professional associations like the American Cancer Society or American Heart Association.
Morality doesn’t exactly correspond to this modern image of efficacy and responsibility (the ethos represented in malpractice suits), but it does align with the idea of a trustworthy physician. Untrustworthy and unethical physicians come out as scandals because it isn’t the expectation of a patient, and Jekyll’s untrustworthy behavior (bleeding-edge experimentation) makes him a poor physician.
I can’t find as clear of a relation for health as the specific nature of Jekyll/Hyde’s addiction and immorality, but it is relatively clear that he is an unhealthy man. Especially near the end, when he and Lanyon become so scarred by the existence of Hyde and Jekyll by the constant transformation that they become pale, sickly, and finally dead. This is clearly an unhealthful state, discrediting Jekyll as a physician if he still has any, and demonstrating the relation between morality and health. This is also encoded by Hyde’s deformity without malformity and his dual status as a very unhealthy man. So there is clearly some sort of link between health and morality and especially physicians’ behavior, but I can’t exactly specify it.