There is a machlokes between the Mechaber and the Rema concerning the berachos recited on the Yom Kippur haftarah by Minchah. The Mechaber says (Shulchan Aruch, Orach Chaim 622:2) that we take the Torah out and read the parshah of arayos and then read Maftir Yonah. He says that we recite the berachos of the haftarah before and after the haftarah. If Yom Kippur falls out on Shabbos, we mention Shabbos in the berachos. The Rema argues that we do not recite the berachah of “al haTorah v’al ha’avodah” by Minchah.
The Vilna Gaon explains that this machlokes is based on a fundamental difference of opinion as to the nature of why we read the Torah and haftarah at Minchah on Yom Kippur. The Mechaber holds that the reason why we read the Torah and a haftarah at Minchah on Yom Kippur is because it is part of the service of Yom Kippur. This is similar to the fact that we read from the Torah on Shabbos by Minchah. Therefore the Mechaber rules that the berachah of “al haTorah…” is recited, like it is recited on Shabbos. And if Yom Kippur falls out on Shabbos, we mention Shabbos in the berachah.
The Rema’s view is that the Torah reading at Minchah on Yom Kippur is unrelated to the kedushah of Yom Kippur; rather, we read the Torah and the haftarah because Yom Kippur is a fast day, and on all fast days we read the Torah and haftarah at Minchah. Even though on a general fast day the reading of the Torah is from the parshah of “Vayechal Moshe…” the specific reading may be changed. On a regular fast day we do not recite the berachah of “al haTorah”; therefore the Rema rules that we should not recite that berachah on Yom Kippur.
A halachic ramification should result from this machlokes. Take this scenario, for example: If a man is sick on Yom Kippur and must eat, may he receive an aliyah by Minchah? If the essence of why we read the Torah by Minchah on Yom Kippur results from the kedushah of Yom Kippur, this sick man may receive an aliyah since he has not violated the kedushah of the day. But if the reason why we read the Torah is because Yom Kippur is a fast day and on fast days we read the Torah by Minchah, then he would not be able to receive an aliyah, as he is not currently fasting. As the Shulchan Aruch, Orach Chaim 566:6 rules: one who is not fasting may not receive an aliyah on a fast day.
Rabbi Akiva Eiger, in Teshuvos 24, rules that one who must eat for medical reasons may receive an aliyah at Shacharis on Yom Kippur. This is because the Torah reading at Shacharis definitely stems from the kedushas hayom. However, he says that he is unsure if he may receive one at Minchah, for perhaps that Torah reading results from the fact that it is a fast day.
Based on the Vilna Gaon’s explanation, this matter should depend on the machlokes between the Rema and the Mechaber. According to the Mechaber one should be able to receive an aliyah even if he is not fasting. According to the Rema he should not be able to receive an aliyah.
The difference between whether the Torah reading results from the kedushas hayom or if it results from the fact that it is a fast day is a fundamental difference in the essence of the Torah reading. If it results from the kedushas hayom, it is a regular Torah reading that is essentially public study of Torah. If we read the Torah because it is a fast day, then the purpose of the reading is essentially to rebuke the congregation. The main rebuke is actually found in the haftarah, except that we cannot read a haftarah only without reading from the actual Torah first. This is the understanding behind the opinion in Megillah 22b that says that only one aliyah is required on a fast day, since the main purpose is to reach the rebuke found in the haftarah.
There are two different sources for why we read the parshah of the arayos at Minchah on Yom Kippur. Rashi, in Megillah 31a, says that it is because one who has committed the sins of arayos should do teshuvah. Similarly Tosafos there says that we read that parshah because women dress nicely on Yom Tov and thus we want to remind everyone of the possible aveiros about which they should be cautious.