Impeachment is analogous to indictment in regular court proceedings; trial by the other house is analogous to the trial before judge and jury in regular courts. Typically, the lower house of the legislature impeaches the official and the upper house conducts the trial.
At the federal level of government, agencies such as the U.S. Department of Justice's Civil Rights Division, the U.S. Department of Education's Office for Civil Rights, and the Equal Employment Opportunity Commission (EEOC) investigate claims of civil rights violations and discrimination.
Government is the main provider of justice and fairness in American society. Many government policies and government institutions are explicitly designed to promote these important public values. The most obvious manifestation of this is the criminal and civil justice system. It is the primary way we as a society ensure that criminals are punished and that wrongs are righted. This kind of legal justice is not something that can be reliably provided by the private sector.