Orthodox rabbis involved in civil rights activism did indeed rely on biblical and rabbinic literature, and referred to them when backing their positions. Here are a few of those sources:
Avot 3:18
הוא היה אומר חביב אדם שנברא בצלם חבה יתירה נודעת לו שנברא בצלם שנאמר בצלם אלהים עשה את האדם
[Rabbi Akiva] used to say: Beloved is man, for he was created in the image [of God]; an even greater expression of love is that it was made known to him that he was created in the image [of God], as it is stated: "For in the image of God He made man." (Genesis 9:6)
Rabbi Ahron Soloveitchik, who publicly spoke and wrote in favor of civil rights in the 1960s, made the following connection to modern times:
From the standpoint of the Torah, there can be no distinction between one human being
and another on the basis of race or color. Any discrimination shown to a human being on
account of the color of his or her skin constitutes loathsome barbarity...[Avot 3:18] implies that every human being, regardless of religion, race, origin or creed is endowed with Divine dignity. Consequently, all people are to be treated with equal respect and dignity. [Logic of the Heart, Logic of the Mind]
The mishnah in Avot is associated with halakhic principals known as:
Human Dignity and Righteousness
Derived from Genesis 1:27 ("In the image of God He created Man"), kavod habriyot (human dignity) is a concept discussed throughout talmudic literature regarding legal exemptions based on undignified situations. Rabbi Ahron Soloveitchik writes that alongside the concept of tzedek (righteousness), it is the Jewish cornerstone of civil rights. He refers to a discussion in the Jerusalem Talmud (Bava Metzia 2:5), in which Rabbi Shimon ben Shetach returned a pearl to an Arab pagan, after the pearl was discovered on the donkey he acquired from the Arab. Although it would not technically be considered stealing, Rabbi Shimon ben Shetach argued:
מה אתון סברין שמעון בן שטח ברברין הוה. בעי הוה שמעון בן שטח משמע בריך אלההון דיהודאי מאגר כל הדין עלמא
“Do you think that I am such a barbarian? I am more interested in
hearing the exclamation, 'Blessed be the God of the Jews' from the mouths of pagans than
I am in making a living.”
Here is how Rabbi Soloveitchik understands this Yerushalmi:
In this story, Shimon ben Shetach gives a remarkable definition of the term
“barbarian.” According to him, anyone who fails to apply a uniform standard
of mishpat, justice, and tzedek, righteousness, to all human beings regardless of origin, color, or creed is deemed barbaric.
From this Yerushalmi, coupled with the concept of k’vod habriyos, one must assume that
those people who refuse to grant any human being the same degree of respect that they
offer to their own race or nationality are adopting a barbaric attitude.
The concept of pursuing justice and righteousness, of course, is based on "Justice, justice, you shall pursue" (Deuteronomy 16:20) and other biblical injunctions to form courts maintaining justice in society.
As mentioned in the comments, many of the Jewish activists during the civil rights movement came from non-Orthodox denominations and may not have relied on or accurately depicted traditional Jewish texts in defense of their actions. We do know, however, that many Orthodox leaders and institutions were in favor of social change, and fought for civil rights based on Jewish principals. See this link for the work of other Orthodox leaders in the 1960s.
