3 scenarios:
1) It's over. He's gone. He will no longer be contacting you. You take note of this and pass the information along to your child. Both of you begin to grieve the relationship. You pick up the pieces and begin to draw boundaries against his return. In time, you will heal.
2) He's not gone. He's just being passive aggressive against you but is using your child's Birthday to hurt you- with little thought given to your child. He decides to take out his aggression on you in a form of passive punishment against you while triangulating
(read definition) your child. You are now a persecutor and he is a victim- your child is also a victim that he can rescue. He has every intention to have your child side with him (once you contact him) to save your child from being victimized by this behavior. The blame will be yours that he missed the Birthday. Your child will not understand this and will unite with him against a common enemy- you.
3) He's not gone but he's filled with hatred for both you and your child. His silent treatment is directed at punishing the both of you. Your child will not understand this and will worry that somehow, they got punished for nothing. When you contact him again, your child will not understand how to resolve this inner conflict and will likely avoid both of you because of confusion.
I think #1 has the best opportunity for advancement out of crazy acres.