Can an employee get terminated with a simple mistake in his paperwork?

My son was working in a hospital in the Bronx, New York, while in training the person that was training him was very nasty and just snatching things out of his hand and speaking to him in a way that no person getting trained should be spoken to. He was calling him by saying "hey Puertorican come here", "hey man", "hey boy", but never by his name in front of all his co-workers, the person also training him was talking about him in front of all his co-workers and in front of my son saying he was not doing a good job, everyone that worked with him was talking about it. The supervisors got a hold on what was going on and went up to my son and told him they heard what was going on and that the person that was training him is always like that and that is the kind of person he is if any problems to let them know, this was on a Friday, that same Friday when my son went home he received a call saying not to come to work after he was on the schedule to work, Monday my son shows up to work and was told to go to HR because of a problem with his paperwork, he goes and gets terminated because of a question he answered mistakenly wrong and was told he falsified his application without giving a chance to explain and after a week he was hired. I am not sure if this coincidence but is this right or at least is the person that was training him was right for saying and doing the things he did? My son did record the person training him just in case.