א. וַיֵּרָא אֵלָיו יְ־הֹוָ־ה בְּאֵלֹנֵי מַמְרֵא וְהוּא יֹשֵׁב פֶּתַח הָאֹהֶל כְּחֹם הַיּוֹם:
The obvious question is why the third day and not earlier. This is the basic source of the mitzva of Bikur Cholim. One of the explanations, quoted by Rashi, says that this was on the third day because that is the day that the pain is worst and the real healing starts. However, this still does not explain why Hashem waited until then and did not pay the visit earlier. One possibility is based on the fact that this was the day that Avraham actually was able to sit "in the doorway of the tent" and receive visitors. Even though the pain was not as bad earlier, he was still suffering the trauma of the surgery and was recovering from the shock. That is why a person is placed in a recovery room after surgery and requires a certain amount of time before visitors are allowed.
Only when he was able to go to "the door of the tent" was he able to receive visitors. Thus, it is improper for someone to actually visit a sick person when it would be more debilitating for them to receive visitors. Notice also that Hashem does not speak to Avraham or tell him anything. It is up to the sick person to decide how much he wants to talk and what he wants to talk about. It is only when Avraham actually gets up and addresses the "messengers" that interaction is initiated.
We should learn from this how to behave when visiting a sick person, just as we learn a similar lesson in how to behave when paying a shiva call.