Is there a law that states nursing homes have to take action against aggressive residents?

I don't have enough information about the incident you are questioning to fully answer this question.  Additionally, the answer would in many ways depend on the state you are in and that particular state's laws regarding regulation of nursing homes.  That being said, all facilities and healthcare providers are responsible for providing a safe environment for their patients, residents and visitors.  If the facility is aware of a potentially dangerous situation and fails to take measures to correct the condition, they could be found to be negligent and held responsible for any damages which may arise as a result of the dangerous condition.