This isn't really childish, he might not be comfortable with you or possibly unwilling to talk because he hasn't grown into the relationship properly.
You "pester" him with your feelings? Well, at least you're honest.
Sounds like you're the childish one.
Being compatible in how you deal with conflict is important in a relationship. It can be a deal breaker if you're not compatible in this area.
Are you incompatible in any other important areas?
Such as your attitudes to money?
He's not necessarily childish, he just deals with situations differently from you. You both have different personalities and this clashes badly when you guys have arguments or whatever. Guys also have a tendency to be scared of anything emotional and their response is to walk out of the room, something like that to distance themselves. Lord knows whenever I fought with my ex he'd storm out and smoke a cigarette instead of talking.
I know it's frustrating because you want to get everything out in the open right there and then but you need to think about how to approach him in a way he'd be more receptive to. If he wants time alone you could say 'let's discuss this when you feel ready' and leave it at that. I also find that people will have their defence up if you come at them in an angry or forceful or 'pestering' manner, so when you do discuss it make sure you come off as calm and not pressuring him. During such a time you could say to him nicely and calmly that you feel like you would appreciate more communication.
If it still doesn't change after that then I'd say you'd have to just accept it as a personality clash and either choose to leave or stay.
the lack of self-awareness is strong with op