Dogs are family members. You feel sad when your normally family does, why wouldn't you feel the same way when the member of the family that gives you unconditional love dies as well?
I'm sorry that you had to put down the dog. It's always hard, but I think it was a wrong choice to not be there (and too many people do it so don't feel bad, I'm not blaming you).
I've been told a lot of stories from friends that are vets about families that rightfully couldn't take the pain of watching their pets be put to sleep, I get it, but my friends always tell me they wish there families could stay. The reason being that their pets get really scared and they spend their last moments without their families at their sides, in a strange place, and in fear. I implore everyone, when the time finally comes that you need to put down your pet, I know it's hard cause I've had to do it, but please please please be strong for them, no matter how painful it is for you, it's a million times scarier for them to be without you at that moment. They spent their lives being there for you, in their last moments they need you to be there for them.
It's going to hurt from a while op, but the pain will pass, your dog will always be in your heart and you'll be able to get another dog who will be part of the family as well.