It is a Torah commandment to set up a beis din, a Jewish court, as it says, “Judges… you shall appoint in all your cities… and they shall judge the people with righteous judgment” (Devarim 17:10). Disputes must be resolved in a beis din, where knowledgeable rabbis are the judges. It is forbidden to resolve a dispute in a civil court, as it says, “These are the ordinances which you shall place before them” (Shemos 21:1), which Rashi explains (from Gittin 88b) “before them, [the beis din] and not before a gentile tribunal.”
One obvious reason is that Jews have a unique legal system, and a civil judgment does not necessarily comply with Jewish law. This injunction, however, applies even if the civil court is committed to rule according to Jewish law. The reason being that a civil court disregards the supremacy of G-d and His law, and as such, it would constitute a desecration of G-d’s Name to bring one’s case there.
The salient distinction is that civil law is secular and Jewish law is G-dly. The basis of civil law is society’s compact for the purpose of maintaining law and order, each country enacting laws according to its fundamental values. Jewish law, by contrast, is a combination of civil and religious law, e.g., if one steals, he is simultaneously sinning against the victim and against G-d.
The Lady Justice statue seen in many civil courts, blindfolded, with a sword in one hand and scales in the other, originates in the Greek and Roman mythic goddess, Themis. Jewish law, on the other hand, is permeated with the consciousness of “for the judgment is G-d’s” (Devarim 1:17).