It’s hard to know who has the problem from what you’ve written. Apparently it’s a cycle. A cycle has no beginning and no end.
We could say: You get snippy. He gets mad and shuts down. You push for intimacy. He wants space. You get snippy. He gets madder. . . We could just as easily say: He gets mad and shuts down. You get snippy. He shuts down more. You push to get close and resolve things. He wants space. . . Around and around it goes. You both have a part in this.
Establishing who starts this kind of thing is fruitless. Neither of you is to blame. Both of you need to talk about how your own patterns get hooked into the cycle and what you can do together to call a halt and do things differently. Some couples can figure this out on their own. Others find it helpful to see a counselor to help them learn new ways to react and to support each other when things are tense.
This is a new relationship. Part of becoming a couple is figuring out how to manage conflict as a team instead of blaming yourselves or each other. Part of loving someone and developing intimacy is learning how to go through the tough moments with mutual respect. It’s a difficult project but when a couple is successful they deepen their trust and love for each other.
I wish you well.