Great question! In case you are in a hurry and don't have time to read everything I researched on the topic, I invite you to skip to section 6 below, which is where I feel your question is best addressed.

1) The Rosh's Kashya

I see the Mefarshim ask a different but related question, namely, rather than learn from Moshe's words by the episode of Korach (Bamidbar 16:29), why doesn't Reish Lakish learn from the verse, "Yeilchu Vah" (Shemos 18:20), since that is where Chazal elsewhere (Bava Kama 100a; Bava Metzia 30b) derive the obligation of Bikur Cholim? See the Perush ha'Rosh (Nedarim 39b, DH l'Bikur).

2) Differentiating from Your Question

Your question comes from a different angle, based on another statement of Chazal (Bava Metzia 86b) in which Hash-m is understood to have visited Avraham Avinu after his Bris Milah (Bereishis 18:1).

3) The Rosh's own Terutz

The Rosh, in any event, suggests the following answer, which some consider as being difficult to accept (see section 4 below). He writes that the reason is because the Pasuk of "Yeilchu Vah" is not explicit , but the reference in the verse by Korach is explicit. The Rosh himself notes the difficulty with this in terms of Reish Lakish's language in the Gemara, which is "Remez." The Rosh evidently understands that the term "Remez" means an explicit reference.

4) Maharatz Chayos

The Maharatz Chayos (Nedarim 39b) takes issue with this, though, because we find other examples (which he lists there) of what Chazal refer to as a "Remez" to various laws even though they are not at all explicit. (By the way, he also notes the significant discussion between the Rambam and Ramban (Sefer ha'Mitzvos, Shoresh 3) about whether the status of laws based on Remez is that of a Torah level.)

5) Two Suggestions

In terms of your question, though, it would seem to me at this point that there are two possible initial thoughts even if they are not full answers.

a) Explicit versus Implicit

The first thought would be, if we accept the Rosh's view above, to say that arguably it is more explicit when Moshe says that people visit one another when deathly ill (as the Tosfos and Ran explain Pshat in our Gemara), and less explicit when Hash-m is interpreted as having appeared to Avraham as a measure of Bikur Cholim after his Milah, since the latter is not openly stated in the Torah but rather inferred. But in a sense, this would be difficult to suggest, since -- as the Maharsha points out -- the literal context of the Pasuk by Korach is not really Bikur Cholim. Rather, as I would understand it to mean, it's referring to the hypothetical option of "if the fate of all mankind befalls them (i.e. natural death)." We interpret the phrase to refer to visiting the sick only because of a Derashah of Chazal.

b) Emulating G-d versus Personal Obligation

A second thought would be this: The story of visiting Avraham only teaches us that visiting the sick is a worthy -- and even Divine -- activity, but not necessarily is it a unique Mitzvah unto itself. This, however, also may not be entirely convincing, because we are in fact obligated to emulate G-d's ways, concerning which Chazal indeed cite Bikur Cholim as a prime example (Sotah 14a, based on Avraham)!

6) Torah Mitzvah versus Rabbinic Mitzvah

a) The two suggestions above, therefore, to my mind, are speculative and not well enough substantiated, when compared to the following which I personally find to be the most appealing answer.

b) The Perush of Rabeinu Avraham Min ha'Har explains that Reish Lakish is not seeking the source that there is a Mitzvah d'Oraisa to visit the sick. Rather, Bikur Cholim is only a Mitzvah d'Rabanan, which Chazal instructed us to perform, because it involves the Jewish values of kindness and caring for a person in need, etc. Accordingly, our Gemara is not at all trying to locate a Pasuk which teaches us that it is a proper activity to visit the sick. (If it were, then as you asked, we should prove it from Hash-m visiting Avraham.)

c) Rather, the Gemara is trying to show that this practice was something that Jews regularly performed, even without being obligated by a Takanah of Chazal. If so, then we can understand very well why the Gemara chose to cite Moshe's speech by Korach rather than the case of Hash-m visiting Avraham. For the former expresses the fact that Bikur Cholim is an activity that Jewish people regularly do; whereas the latter, while it's certainly a noble act of benevolence, to be sure, nevertheless it is an act performed by ha'Kadosh Baruch Hu, not people in the Jewish community.

7) The Rambam

By the way, the Rambam (Hilchos Avel 14:1) adopts the view that the Chiyuv to visit the sick is mid'Rabanan, but still it is included in the Torah obligation of "Love your fellow as yourself" (Vayikra 19:18). It might be argued that this is a case of the phenomenon -- which appears elsewhere as well -- of a Chiyuv Mitzvah that is mid'Rabanan (Bikur Cholim) which involves a Kiyum Mitzvah that is mid'Oraisa (v'Ahavta l'Re'acha Kamocha).

I hope this helps!

Warmest regards,

Yishai Rasowsky