Should an organization terminate the employment of an employee if he/she violates an AUP? Why?

Not all violations of AUPs cause the same damage, and hence cannot have the same corrective and punitive actions. The consequences would depend on the type of violation, cause of the violation, effects of the violation, history of the violator (first-time violator/repeat offender). The corrective action could be a warning, a write-up, or a termination, and even a lawsuit. There is no one standard consequence unless the policy explicitly states so.

More often than not, the consequence is decided by the immediate manager, management team, and possibly the legal team, on a case by case basis.