Why does the Torah begin with ֻןיְרִיב וְכִי ִׁיםֲנָשא, that there was a fight? It would be sufficient to write ֵהוֵּער ֶתא ִׁישא ִכָּהוְה, that someone hit his fellow man, unrelated to whether there was a fight beforehand or not.
Chasam Sofer answers that the pasuk begins with a dispute to hint that doctors are needed solely when there are fights and conflicts. When there is peace among people, no doctor is required. Hashem heals. This is alluded to in the pasuk (Yeshayah 57:19) ָלוֹםׁשִיוָאתוּרְפ 'ה ַרָמא ָּרוֹבַקוְל ָחוֹקָרל ָלוֹםׁש, "Peace, peace to the far and to the near," says Hashem, "and I will heal him." When people are at peace with those close to him and with those far from him, he is at peace with everyone, ַרָמאִיוָאתוּרְפ 'ה, Hashem will heal him. He doesn't have to go to the doctor.
On the pasuk (Shemos 15:26) ֶָךְארֹפ 'ה ֲנִיא כִּי, "For I, Hashem, heal you", the Baal HaTurim writes that this time, the letter 'פ (letter feh) of ֶָךְארֹפ is written without a dot in the 'פ, and it is thus read softly. In contrast, in the pasukֵּאַפיְר ֹּאוְרפ, there is a dot in the letter peh, and it is read with more force. This is because "When the refuah is from Heaven, it comes easily. When the healing is from man, it is hard and painful, and therefore, the wordsֵּאַפיְר ֹּאַפוְר are written in parashas Mishpatim with a dot in the peh."
The Gemara (Horiyos 14.) says there was a question about who should be the rosh yeshiva, Rabba or Rav Yosef. Rav Yosef had vast knowledge. He knew all the Mishnayos and all the Braysos. (He was called סיני because all the Mishnayos were organized in his mind, as they were given from Har Sinai.) Rabba's excellence was in his logic, svarah, and pilpul. (He was called הרים עוקר, uprooting mountains, implying that he uprooted mountains with his logic.) The question was which type of expertise was more important for leading the yeshiva.
They sent their doubt to the rabbanim in Eretz Yisrael. The rabbanim in Eretz Yisrael responded that expertise in bekiyus, to know all Mishnayos and Braysos was a more important quality needed to lead a yeshiva. Nevertheless, to honor Rabba, Rav Yosef refused to lead the yeshiva. For the next twenty-two years, Rabba led the yeshiva. (When Rabba was niftar, Rav Yosef took over the yeshiva.) During those twenty-two years, Rav Yosef showed no sign of leadership and power. He didn't even call a doctor to his home. Rashi writes that when he needed a doctor, the doctor came to Rabba's home and treated Rav Yosef there. He wanted it to be clearly known that he wasn't the rosh yeshiva.
The Tosfos Rosh writes in the name of the ה"רמ another reason Rav Yosef didn't call a doctor to his home. It is because he and his family were well and didn't need a doctor. For the twenty-two years that Rav Yosef humbly gave over the yeshiva to Rabba (because he didn't want to lead the yeshiva when Rabba could), he was blessed, and no one in his family became ill. The Rosh concludes, "This explanation seems correct to me."
Reb Chaim Palagi (Refuah v'Chaim 12:74) writes, "When there is an ill person in your home, practice vatranus, even with the birds of the heaven. It is tested and proven that they will pray for you."
Notes
14. Sometimes a person thinks, "I should be mevater for that person?! He is like an animal in the form of a human being." Therefore, we are taught to be mevater even for birds. Even when someone bothers you like a bird dancing on your head... even then be mevater, and they will daven for you.