Correct grammar after an argument?

Just because you've lost an argument doesn't mean you need to be a child about it."

Perfectly acceptable grammatically.

The structure is:
Just because you (did this), it does not mean you need to (do that)

Perhaps you would have understood:
Just because you've lost an argument doesn't mean you need to act like a child.

"

"Childish"...usually comes into play...but I guess the way it was put is okay also.

"Just because you've lost an argument doesn't mean you need to be a childish about it"...sounds better to me, but honestly, I cannot see a big difference.

In Britain we have handy phrases for this kind of behaviour; "Don't throw your toys out of the pram", or "Don't spit your dummy out" (comforter)