Even if you were to text his ex to ask her to respect your relationship, that may actually cause her to do the opposite and send even more Facebook and Instagram messages to your boyfriend.
In essence, once she notices that the situation has reached this point, she may get the feeling that she may have even more of a shot to win him back than she thought. Hence, she starts trying even harder.
If you feel that he has gotten over her, it may be best to put out your efforts in making your relationship with him the best that it can be.
Avoid falling into the trap of becoming vocally suspicious of what he may or may not be doing with this ex behind your back. This would only cause a rift between the both of you that could easily lead him into the arms of his ex.
Focus more so on always having open conversations with your boyfriend. Always make him feel that he can come to you with any issue.
As long as he feels that he can come to you with any issue and talk openly without the fear of being judged, he will have no reason to go to his ex for any kind of emotional comfort or support.
His ex will most likely eventually stop sending him these Facebook and Instagram messages when she notices that he is continuing to ignore them and she is not having any emotional influence on him.