Milah heals the soul by removing concealment

וירא אליו ה' פרש"י ובמדרש תנחומא לבקר את החולה.

"And Hashem appeared to him" (Bereishis 18:1) — Rashi and the Midrash Tanchuma [explain: He came] to visit the sick.

Hashem appeared to Avraham to perform bikur cholim, visiting him as he recovered from his bris milah.

וכן אמרו בגמ' שקבעו רפואה בשמינית בתפלה משום שניתנה מילה בשמיני שצריכה רפואה.

And likewise they said in the Gemara that they fixed [the blessing of] healing (refuah) as the eighth blessing in the Shemoneh Esrei, because milah was given on the eighth [day] and requires healing (Megillah 17b).

The placement of the berachah for healing as the eighth in Shemoneh Esrei is tied to milah on the eighth day, which calls for refuah — linking the eighth, milah, and healing.

והענין הוא כמ"ש ראו עתה כי אני אני הוא אני אמית ואחי' מחצתי ואני ארפא.

And the matter is as it is written: "See now that I, I am He; I put to death and I bring to life, I have wounded and I will heal" (Devarim 32:39).

The verse pairs death with life and wounding with healing — Hashem's bringing of life and healing comes specifically through a prior "death" or "wounding."

ואיתא בזוה"ק שלח אני אמית לסט"א ועי"ז מחי' סטרא דקדושה וע"ז כתיב ה' ממית ומחי' ע"ש.

And it is brought in the Zohar Hakadosh (Shelach): "I put to death" — to the sitra achra (the side of impurity), and through this He gives life to the sitra di-kdushah (the side of holiness); and regarding this it is written "Hashem puts to death and gives life" (Shmuel I 2:6), see there.

Hashem "puts to death" the forces of evil, and by that very act He vivifies the realm of holiness — death and life are two sides of one act.

וכן הוא במילה שנתן הקב"ה לאברהם אבינו ולזרעו להמית כח הסט"א בערלה ועי"ז מחי' הנפש כי הנפש אינה יכולה להתגלות בגוף מכח הערלה ולכן היא חולת אהבה וע"י המילה מתרפאת וכמו שיהי' לעתיד בכלל שיעבור כל הסט"א ואז יראה כבוד מלכותו כי אני הוא ולית שולטנא אחרא כלל.

And so it is with milah, which Hashem gave to Avraham Avinu and his offspring — to put to death the power of the sitra achra in the orlah (foreskin), and through this to give life to the soul; for the soul cannot be revealed in the body because of the orlah, and therefore it is "lovesick (cholas ahavah)," and through milah it is healed — just as it will be in the future in general, when all the sitra achra will pass away, and then the glory of His kingship will be seen, "for I am He, and there is no other dominion at all."

Milah enacts this same principle: it "kills" the impure power lodged in the orlah and thereby gives life to the soul, which until then was concealed and "lovesick," unable to shine through the body. This is a foretaste of the future, when all evil will be removed and Hashem's sole sovereignty will be revealed.

ומעין זה בפרט איש ישראל אחר המילה וירא אליו שנתגלה הקדושה.

And something of this occurs in particular to a Jew after milah — "and [Hashem] appeared to him" — that the holiness is revealed.

On an individual level, after a Jew's milah the divine holiness becomes revealed, echoing "Hashem appeared to him."

מחצתי ואני ארפא אחז"ל מחיצה שעשיתי אני מרפא וכמו כן הערלה מחיצה כמ"ש וירא אור כי טוב וגנזו לצדיקים.

"I have wounded (machatzti) and I will heal" — Chazal expounded: "the partition (mechitzah) that I made, I [Myself] will heal"; and likewise the orlah is a partition, as it is written "and He saw the light, that it was good" (Bereishis 1:4) — and He hid it away for the tzaddikim.

Reading "machatzti" as "mechitzah" (partition), Chazal teach that Hashem heals the very barrier He made. The orlah is just such a barrier, concealing the holy "hidden light" of creation that was stored away for the righteous.

ולכן הוא בשמיני שעיקר הרפואה בביטול הטבע ואז מתגלה אור שלמעלה מהטבע מעין מ"ש ושב ורפא לו שעיקר הרפואה בא בהתקשרות אל השורש שהוא בחי' תשובה:

Therefore it is on the eighth [day], for the essence of healing lies in the nullification of nature (bittul ha-teva), and then the light that is above nature is revealed — akin to what is written "and he will return (ve-shav) and be healed for him" (Yeshayahu 6:10) — for the essence of healing comes through binding oneself to the root, which is the aspect of teshuvah.

The number eight transcends the natural seven, so milah on the eighth day represents going beyond nature. Real healing comes through "returning" — reconnecting to one's root in Hashem, which is the inner meaning of teshuvah.

Summary: Hashem appeared to Avraham to visit the sick after his milah, and the berachah of healing is the eighth in Shemoneh Esrei because milah is on the eighth day. The principle of "I put to death and bring to life, I wound and heal" is that Hashem gives life to holiness by putting evil to death. So too milah "kills" the impure power in the orlah — the partition that conceals the soul, which until then is "lovesick" and cannot shine through the body — and thereby heals and reveals the soul, a foretaste of the future removal of all evil. Because the eighth day transcends nature, real refuah comes through bittul of nature and binding to one's root, which is teshuvah.