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.
First, civil government is secular in nature while Christianity is spiritual. Christians are aliens and strangers, just passing through this world (see 1 Peter 1:1). Their citizenship is in heaven (Philippians 3:20). Second, the state can look upon Christianity as competitive, even hostile to its authority.