In California. If parents share 50/50 legal and physical custody of a child, can one parent demand that their child not see a particular person while it’s the other parents time?
First of all, there is no such thing as "50/50 . . . custody." Under
California law, there are two different types of custody: physical custody and legal custody. Physical custody is concerned with where the child lives. Legal custody is concerned with decisions relating to the health, education, and welfare of the child. The two types of custody can be either "sole" or "joint." If you click the link above, you'll see what those terms mean in the context of each type of custody.
Second, anyone
can "demand" anything that one wants, so I assume the question you asked isn't what you actually intend to ask. If the question you intended to ask is whether one parent has the legal right to dictate unilaterally whom the child may or may not see during the other parent's custodial time, the answer is no. Any restriction of that sort would have to be made express in the divorce judgment or custody order.
when it’s the moms time with the child she decides to just let him stay with the child’s aunt and basically lives with the aunt on that potion of time.
This is ambiguous. Are you saying that both the child and the mother live with the aunt during the mother's custodial time? Or is the child living with the aunt without the mother there?
The dad . . . his child to stop living with the aunt on the moms time.
What happened when the father spoke with the mother to express his dissatisfaction with this situation?
The aunt is also currently being investigated for child abuse for a child she has in her care.
Does the father have any evidence of or reason to believe that the aunt is committing any sort of abuse against the child?
Could the dad file a restraining order against the aunt so that the mom is forced to keep the boy away?
No. Only a court can file a restraining order. The father could file an application for a restraining order. Whether there's any reasonable chance of the court entering the order is impossible to predict based on information provided.
The mom doesn’t know yet the aunt is being investigated
Why on Earth would the father not convey this information to the mother?