Well, symbols are generally arbitrary by definition, so if someone adopted the caduceus as a symbol for medical services or medicine in general, it works as such a symbol in a given environment, as it does in modern USA.

The association of the snakes with medicine in Europe also dates back to Ancient Greece, especially to the cult of Asclepios and Hygeia that eventually led to the adoption of the serpent coiled around the staff and the chalice as the symbol of medicine and pharmacy respectively. In the case of these two deities, the association with health and healing is undisputed and thus the link between the Asclepios' staff and medical services seems to be much stronger than with other symbols derived from the ancient mythology and customs. The usage of the Graeco-Roman symbols waned in the Middle Ages (some argue that one of the most common attributes of a physician even until early 17th century was a flask, either for examination of urine or for holding medicines) but it re-surged in the 16th century with the revival of the ancient tropes in art and literature, where it flourished until this day.

The adoption of the caduceus in the context of medicine in USA is often attributed to Cpt. Frederick Reynolds of the U.S. Army Medical Corps, who proposed the usage of this symbol arguing, that it is common among European forces and thus fitting for the usage in USA. This proposal, eventually accepted by U.S. Army Surgeon General William Forwood in 1902 was likely based on the erroneous name attribution, as European armies commonly used Rod od Asclepios or red cross, and the symbol is described by Reynolds as the former. So it is most likely a case of a mistaken identity on the proponent's part. Some argue that the error might have been caused by John Churchill, London publisher who had a large portfolio of medical works and used the caduceus as the printer's device (trademark) of his publishing house since late 19th century. It is worth noting however, that this symbol was quite popular among printers since 16th century, quite fittingly, as the distribution of books was largely considered a type of communication what explains the usage of Mercury's attribute.

It is worth noting, however, that William Emerson in Encyclopedia of United States Army Insignia and Uniforms states that the symbol was adopted by the U.S. Army Medical Corps deliberately, as it symbolizes not the medical mission of the Corps but rather its neutrality and non-combatant status, and thus in military context the usage of caduceus is logical and appropriate.

Now, is caduceus a good symbol of medicine from the logical standpoint? Not really. The main reason is that its shape is by no means incidental and it carries a certain message that is related not as much to the god Hermes, but rather to a specific mythical event. To wit, the two snakes intertwined and facing each other while being coiled around the winged rod are a visual reference to one of the mythical stories about Hermes who, upon seeing two fighting snakes, threw his staff between them, and the animals coiled around it and around themselves, immediately forgetting their animosity. Thus, the caduceus was first and foremost the symbol of peacemaking or diplomacy and was used in Classical Greece in precisely this capacity, being the symbol of keryx, a diplomatic envoy. Of course, as in the case of many stories derived from the mythology, the origins of the symbol is definitely obscure, and is is posited that the caduceus or even the figure of Hermes itself might be an adoption from much older beliefs, sometimes related to the Sumerian mythology.

In other words, as an attribute of Hermes, caduceus is no better symbol for medicine than winged sandals of winged cap, commonly associated with this deity. In addition, link between Hermes and medicine is very tenuous at best if not contradictory, as one of the aspects of Hermes/Mercury was that of a psychopomp, i.e. a party helping in the passage to the underworld what in the context of medicine overlaps with a definitely morbid sense of humour. Thus, it is largely accepted that symbols associated with Asclepios, Hygeia or Panacea, figures tied primarily to the field of medicine, are more 'appropriate' to symbolize medicine and related fields.