Generally speaking, when people think about fasting in terms of Torah, they are thinking about the type of fasting associated to the 9th of Av or Yom Kippur, meaning neither eating nor drinking. This is related to the idea of rising above the physical/material plane of existence.
In the context of a one day fast, or even a two day fast, that is possible, but is also a severe strain on the body.
However, in the context of making a tikkun over past transgressions, the number of fasts prescribed in Torah is often many more days than simply a day or two. See for example the references in the Tanya in Igeret HaTeshuvah at the end of chapter 1 to Sefer Rokeach and Mishnat Chassidim and also in chapter 3 there.
Medically speaking, fasting, meaning abstinence from eating, is something that can be done for many, many days consecutively. The longest consecutive fast from eating on record is over a year. While a total fast, abstinence from both eating and drinking can only be survived three to five days. The abstinence from water is the critical factor.
Does the Torah make a distinction between types of fasts? The Aramaic word for fast is תענית from the Hebrew ענה. The Hebrew root has a connotation of to respond or to answer (as in answering for improper actions) and also to deprive, or to delay, or to submit/become humble or to become poor. All of these relate to the idea of depriving oneself of food in humility and an act of submission to G-d's will.
The consequence being that one burns their excess body fat, which is compared to burning the fat of an offering on the altar brought in the Temple, like is mentioned in the Tanya for example.
So the focus seems to be about the burning of body fat, which relates to the eating of food only.