Combating Racist Training in the Military Act of 2021 This bill prohibits the Armed Forces and academic institutions of the Department of Defense from promoting specified anti-American and racist theories (e.g., that any race is inherently superior or inferior to any other race). Specifically, the bill prohibits including anti-American and racist theories or materials in curricula, reading lists, seminars, workshops, trainings, or other educational or professional settings in a manner that could appear as sponsorship, approval, or endorsement; contracting with, hiring, or otherwise engaging speakers, consultants, diversity trainers, and other persons for the purpose of advocating anti-American and racist theories; compelling members of the Armed Forces to affirm or profess belief in anti-American and racist theories; and segregating members of the Armed Forces by race in any setting, including educational and training sessions.