During an argument my significant other told our kids what they are getting for Christmas. Because I'm this I'm going to shut down and ignore him.
Can you tell me why this is a thing you want to do to hi?
He let me down, he wasn't supposed to tell them anything!
It is very wrong to tell the kids what they are getting for Christmas, but it was an action made in haste. He is at fault, but ignoring him now is something that is very immature. You need to talk it over and then get the kids something else so they will be surprised.
So I don't get to hurt him back? Damn.
Hurting him back is petty and cutting him off won't help resolve the situation and will probably make things worse. I'm sure they feel bad for telling your kids about their presents, but it's something you two should discuss and move past from.