You would need to check with not only the state medical board but the state veterinary medical board/state veterinary practice act. I have not looked at all 50 practice acts in detail, and there is some variation in what licensed veterinary technicians and lay employees can do, but the common theme is that diagnosing, prescribing, and surgery are reserved for veterinarians licensed in that state, and sometimes for visiting veterinarians from other states or students under direct supervision. I would be interested to hear if the veterinary practice act is different in your state (most of them are available online).
As others stated above, there are myriad possible complications. Physicians are not taught comparative medicine across species (and frankly there isn't time in med school from what I understand); while at some level a mammal is a mammal, there are important idiosyncratic differences in physiology and receptor pharmacology that make it difficult to predict based on one species whether another species' response will be the same or entirely different.
Some medications are safe in some species and toxic or otherwise contraindicated in others - acetaminophen is used in dogs but is quite toxic in cats. The seemingly innocuous NPB ophthalmic ointment has an elevated risk of adverse reactions in cats. Benzodiazepines provide great tranquilization/anxiolysis in other species, but are almost never given alone in horses due to the incidence of disinhibition/paradoxical aggression reactions, which can result in the horse injuring itself and people and destroying anything nearby.
Even with the same drug, the dosing may not scale like you think, with dose ranges in mg/kg varying widely between species. E.g., xylazine, commonly used for sedation in large animals, has far greater effects at the same dose in ruminants vs horses.
The most common differentials for certain problems vary by species, and some things that are common in one species might be virtually unheard of in another. How often do MD's see gastric dilatation-volvulus, a surgical emergency which can sometimes present as a dog just looking a little punky and retching occasionally without vomiting?
I'm not denying that an MD is a more informed animal owner than most, but that knowledge should be tempered by humility and acceptance of its limitations and is no replacement for a working relationship with an actual veterinarian.
