There's no real explanation online (not surprisingly) so here's my attempt at it:
The overall theme of "The Prescription" seems to deal with social status and pride of an individual. This is illustrated by Dr. Mehmet's comment where he states,
"‘I decline to accept you as a colleague, Alexander Zarifi.’"
as well as
"‘Your [Alecco's] word is not good enough.’"
Essentially, Mehmet still feels superior to Alecco due to the fact that Alecco was his apprentice, thus lower in social status. But despite the fact that Alecco has reached the same amount of prestige (or perhaps more) disgusts him, so Mehmet gives Alecco a rather curt explanation and attempts to dismiss him off, sort of a "you're wasting my time" attitude. This is portrayed by the following sentence,
"'There is no need for us to waste time in each other’s company.'"
He can't stand to be in the same room (or I guess, outdoor living space?) as Alecco, so quickly dismisses him. It doesn't really work though, as Alecco gives quite a lengthy explanation of why he believes this woman can be cured differently. He starts mentioning his success and studies under different doctors. To this, Mehmet replies,
"'Knowledge is good, but what is the use of knowledge without honesty?'"
This quote in itself doesn't really make sense, so it must be viewed from the overall theme of the story. Now after reading this for a while, I finally understand the implications of this comment. This is a backhanded insult to apprentice's achievements as a doctor. The "knowledge without honesty" is a reference to when Alecco read Mehmet's books without permission. Suddenly, Alecco is filled with guilt (since the reason that Alecco became so successful was due to sneaking and reading the books) so Mehmet is accusing Alecco of obtaining success through lying. This is kind of a jerk move by Mehmet, and we can end the discussion here, saying Alecco's just guilty. But nope! I think there's more, to it than that.
The books has a heavy discussion relating to mania and hypnosis: here, they're referring to traumatic experiences disrupting someone's daily life, perhaps a form of PTSD. I think this is some sort of metaphor/implied meaning. Why else would Alecco go to Vienna to study hypnosis and psychiatrical help if it were not for his own issues? Remember that Alecco nearly did die, and any near-death experience (especially if some poison is shoved down your throat) has got to be traumatic. Also, moving on to the concoction that was supposed to kill him:
"... a dose of aphrodisiac go into the glass, and then the dried flowers of agnus castus, which inhibits sex, opium, lavender, ecballium elaterium, the most violent of all purgatives, datura, either 14 grams, inducing insanity, or 22½ grams (death) and finally mustard and cinnamon."
Funnily enough, most of the poisons that are given a description are pretty accurate ecballium elaterium is indeed a very, very strong purgative, and also poisonous. Datura, too is known for inducing insanity. Agnus castus has historically been believed to be an anaphrodisiac, though the reliability of this is questionable. The other listed are otherwise not harmful, in their natural forms. He might have survived this by excreting all the poison from his system, and would explain his nausea on the ship (loss of water).
TL;DR, the significance of the action is indicative of perhaps PTSD, and the effect of social status and pride of humans.