The active yet nonviolent refusal to comply with morally-objectionable laws, used by abolitionist organizations like the American Anti-Slavery Society and civil rights activists like Martin Luther King, Jr., is known as what?

Civil Disobedience

Explanation:

Civil disobedience is defined as the refusal of a citizen to comply with certain laws or demands of a government. Civil disobedience has to be nonviolent hence the tag "civil".

In the case of Martin Luther King, he promoted the freedom and equality for the oppressed black minority by encouraging civil disobedience of unjust laws such as those banning blacks from certain areas or using certain facilities. He also led protests even when government authority had refused to grant him permission, hence encouraging civil disobedience.