Like you said, childish. What are the arguments about? Do you ever have productive discussions rather than arguments? If you do, what takes them to  the argument level? Either way, once he storms off, the argument/discussion is closed. He doesn't get to call you up with his "and another thing" nonsense.

I don't think I could put up with this for very long ... and I suspect that you feel much the same, right?

So you feel guilty. If he's gonna continually act like a child, he shouldn't be in a relationship. Normal healthy people have a way of communicating with one another.

i don't sugarcoat things for my husband. If skids are being shitty, then i tell husband what ferals they are.

I am someone who needs to take some space after an argument to cool down and think. But I'm talking hours, not a week. And I don't leave the home overnight. 
 

Where does he go? How does the incident end, do you end up apologizing so he will return?

First off, tell him that if he can't have a conversation with you that he doesn't get to text your daughter so he needs to stop that crap ASAP. If he doesn't, block him from her phone. She doesn't need to get dragged into this crap, and yes, talking to her while ignoring you is bringing her into it because that will push your daughter to feel like she needs to say something to you about him. Nope. Don't let him use your kid to get to you.

That actually leads me to my second point. Taking time away, even a week, after an argument to think things through is okay. HOWEVER, there is a right and wrong way to do it.

The RIGHT way to do it, if you're thinking of salvaging a relationship, is to expectations and boundaries first:

"Look, I'm upset and can't talk about it right now. Give me X hours/a day and I'll get back in touch to let you know how I'm processing things and if I am ready to talk."

Then, at that time, they connect with you again to either talk or to schedule a time to talk. Ideally during this time, you'd set parameters on how that talking will go: the topics that will/won't be covered, whether you'll talk in person or over the phone (please not through text), who will get to talk first, and what actions will end the discussion (e.g. namecalling).

You meet, you discuss, and you both decide if you need to talk more or if that cleared things up. If it's not settled, the person who left may decide to stay gone, but again a time is set to check back in. 

What your SO is doing is not helpful, and it is bullying. You can be the one who reaches out first following the formula above if you'd like to move forward on discussion with him. I wouldn't make his leaving the top of the list of things to discuss. "Finish the fight" you two had previously then set aside time to talk about how him leaving hurt you and a better way he can do it in the past if he needs time to himself.

Either he's just bad at communicating and this technique will make a difference, or he doesn't care that he's bad at it so long as he wins. If it's the former, you can make this work (if you want). If it's the latter, then he's not worth keeping around. That's too much volatility.

So, he just leaves in the middle of an argument and stays gone for a week? And comes back when he feels like it? Personally, this would not work for me, as that is not how two adults, living together, handle things. How does any argument/discussion get resolved? 

His excuse, that he does it because he is so hurt, is nothing but manipulation.

Maybe the two of you need a better way to discuss things but him walking out (and staying gone for a week) is not the answer and does nothing for the relationship. Except cause resentment and one day, you will have had enough of this behavior and not want him to come back.

Maybe he's not mature enough for a live-in relationship and should just get his own place.

He is punishing you by absenting himself, and attempting to force you into submission.  Not acceptable behaviour on his part. 

Walking away to cool off so someone doesnt say something ugly is one thing. Leaving for a long period is not ok! I  ved one scene on Grey's Anatomy where a couple was fighting and he grabbed the car keys and yelled "I gotta get out of here" or something like that, but then came back and said in a yelling tone "But I'm coming back becuase I love you!". I loved it! If I were you I would refuse to talk to hi onthe phone or let him talk to your daughter until he comes home and you guys hash things out face-to-face.

You have been together 9 years and he does this? Idk, I gues I would probably pack his stuff and put it on the porch/garage/somewhere for him to pick up. 

Don't text him back. If he wants to talk, he needs to do it face to face like a grown man. 

So he runs from y'all's problems and stays gone as long as it takes for you to miss him and feel guilt for "running him off" with geninue relationship issues so that he can come back and be exactly the same as he was before. Ya know BM used to do something like this with DH. She would run off knowing good and well she would get to waltz back in and he would take on the guilt and responsibility of the argument. The only thing that stopped it was him finally realizing she was playing a game with him trying to "teach him a lesson" aka he had to do/be what she wanted him as she needed him. He changed the locks and moved on. I think you should too. 

That's not healthy. Some people need time to themselves to calm down and think when they are upset and angry. I am one of those people. But I'll go in a different room. Or leave for a half hour. Being gone for that long is not a healthy way to deal with issues. It's immature. You gotta tell him it's unacceptable for him to do that. If he can't deal with arguments without leaving for days than he has issues that need addressed and you should suggest he sees a therapist. Don't accept that abs let it be the norm. 

Sounds like he is grooming your daughter. He's triangulation himself to make your daughter like him more so he can do things. Don't believe me? Happens all the time