My Exh did this. It was utterly horrible, frustrating and turned me into a shrieking fishwife.
Once, he explained to me in a letter what was happening inside his head when he was doing it: he said that because he felt he was being criticised (he always stonewalled during an argument about anything) he would feel a burning shame and violent anger inside and just 'shut down.'
The impact on me and our relationship was god-awful. He would shut down and I would get more and more wound up, sometimes begging him to talk or just say SOMETHING.
Often he would wait until I was just yelling and then he would say quietly 'I can't talk to you when you are in this state' and turn round and walk out. I said to his retreating back so many times 'BUT YOU HAVE GOT ME INTO THIS STATE BY REFUSING TO TALK'
Oh god it was horrid horrid. He would walk away very slowly and deliberately and go and get into bed and hide under the covers usually in the spare room. One night I went in to him and literally begged, crying, for him to talk to me and he told me to 'fuck off'.
His stonewalling was so bad he disappeared on three occasions. Once for two days, a year later for three days and the following year for a full five days. He didn't answer his mobile, I had no idea where he was. On that last occasion, when he walked back in the door, I asked him where he had been and why and he said 'I thought the marriage was over so I just left.' He had gone to see a play in Stratford, partied with mates in Bristol and blown nearly a grand on new clothes.
Stonewalling is very very abusive. The perpetrators can turn it round and make out that it's your fault for driving them away or into silence. But it's totally deliberate, extremely aggressive and very damaging. My Exh was a selfish, self-absorbed, controlling, deceitful liar. He enjoyed seeing me get into such a state while he sat or stood in utter silence. He enjoyed winding me up like clockwork, with his long long silences.
If your partner can use this abusive, selfish, nasty and cruel behaviour even on his daughter's birthday, there is little hope. It is shitty behaviour that he needs to resolve himself - but as someone else said, he won't because inside, secretly, he doesn't think he is ever in the wrong
I think separation sounds a good idea for your little girl's sake and your own mental wellbeing.