When the colonies severed their allegiance to England, in 1776, through the adoption of the Constitution in 1789, they had to have some form of law upon which to deal with matters, both criminal and civil. To do so, they adopted the Common Law of England, as it existed on July 4, 1776. This, then, became the foundation of laws upon which both the federal government and state governments began the process of developing their judicial systems.
What is important to understand is that the laws that they adopted were concerned with Justice. For example, though Webster's 1828 dictionary has no definition of "judicial", an adjective, it does have one for that body that is responsible for that function of government, the Judiciary:
JUDI'CIARY, n. That branch of government which is concerned in the trial and determination of controversies between parties, and of criminal prosecutions; the system of courts of justice in a government. An independent judiciary is the firmest bulwark of freedom.