When our forefather Avraham was recovering from his circumcision, “G-d appeared to him”  in order to fulfil the commandment of ‘visiting the sick’ .
But in seeking a source for the commandment to visit the sick, we are instead taught through the following allusion: When Moshe set out to disprove Korach and his followers, he warned that they would suffer an early death, and only “if they die a natural death like when people visit them first”  would it be clear that they were in the right . Of course they were wrong and they did not die a natural death. But in describing what a natural death entails, Moshe explained “like when people visit them first”, i.e. like people who are sick and are therefore visited. This is where the commandment of ‘visiting the sick’ is alluded to in the Torah.
But what is wrong with learning from the fact that G-d visited Avraham when he was sick? Furthermore, if Korach were to die a natural death, particularly being that he was an adversary of Moshe, the leader of our people, perhaps he would not have visitors as a regular sick person would? Why would a person seek to visit someone like Korach?
R’ Moshe Wittenberg was a rich and learned man, who spent his last years in Jerusalem. R’ Zalman Rivlin, secretary of the Jewish community of Jerusalem, was a steady visitor at R’ Wittenberg’s house, and succeeded in obtaining from R’ Moshe considerable sums for various charitable purposes. However, R’ Rivlin always had to tell R’ Moshe something which would put him in a good humour before he would contribute.
Once R’ Rivlin came to R’ Moshe, and, for once, failed to ask for a donation, saying, that he had merely called to inquire for his health. R’ Moshe was rather surprised at such an unusual call, and said, “Thanks, R’ Rivlin, for calling. I feel fine. But is that all you came to see me about?”
“Well,” replied R’ Rivlin, “King Solomon told me that you are ill with a severe sickness.”
“King Solomon! What do you mean?” asked R’ Moshe incredulously, “and furthermore what kind of sickness do I have?”
“King Solomon says,” answered R’ Rivlin, “there is a severe sickness… namely, riches kept by their owner to his detriment”.
R’ Moshe smiled and replied, “I did not know that I was so gravely ill with this sickness, but thank you anyhow for calling on me, for you have through it performed the meritorious act of visiting the sick.”
“But,” interjected R’ Rivlin, “I have not fulfilled my duty in full yet, as we have been taught, ‘a person who visits the sick removes from him one-sixtieth of the sickness’  – and I have not yet taken from your ‘sickness’ anything at all!”
R’ Moshe understood the hint, and gave him a substantial donation.
We have been taught that in addition to the obvious benefit for the person who is sick, visiting the sick also benefits the visitor: it helps him realise how fickle we are and how uncertain we are of the future. If he takes this to heart he will improve his ways and maximise his time as constructively as possible .
Perhaps this is the ‘sixtieth’ that a person visiting the sick takes away with him. One sixtieth is the measurement in Jewish law where we often apply the concept of ‘annulling’ (one in sixty parts can sometimes be considered as if it does not exist). So too here, visiting the sick might not make him significantly better, but by the same token it acts as a type of ‘immunisation’ for the visitor for him to improve his ways, not requiring him to become sick in order to learn the same lesson directly.
This sheds light on the phrase ‘bikkur cholim’ (‘visiting the sick’). The word ‘bikkur’ is similar to the word ‘boker’ meaning morning, when the sun rises and provides light. A sick person does not always recover, but in light of the potential benefit to the visitor of visiting the sick, there is always a ‘bright’ side to look to: The visitor is ‘healed’ from his ‘sickness’!
Perhaps this is why ‘bikkur cholim’ (visiting the sick) is in the plural: the visitor is also considered ‘sick’ until he improves his ways!
Now we can understand why we do not learn the commandment of visiting the sick from G-d’s visit to Avraham, because whilst there is no doubt that Avraham benefited, Hashem did not. Accordingly, we would never have known that there is also a benefit, or a ‘cure’ to the visitor!
This also sheds light on the rather difficult and controversial teaching that ‘the best of doctors goes to the netherworlds’ . This teaching should of course not be understood literally, and of course sickness is not wished upon anybody. Perhaps though, it is alluding to a good doctor who cures people thereby lowering the necessity to visit the sick, and consequentially the potential visitor loses his opportunity for self-improvement!
Have a bikkur-ful Shabbos,
 Bereishis 18:1
 Rashi, Bereishis 18:1 (see also Sifsei Chachamim)
 Bamidbar 16:29
 Nedraim 39b
 Koheles 5:12
 Kli Yakar, Bamidbar 16:29
 Kiddushin 4:14