What you did was merciful for your beloved dog, I'm sure God does not hold it against you neither does your dog, for he is resting in peace now. You have to forgive yourself of a sin you have not committed. I know it hurts you a lot for we love them so much and they are part of our family, but to put them to sleep is not murder. Murder is killing another human being, that's what the bible means.
Lord help this person to forgive them self for having to put their beloved pet to sleep, and out of his misery and pain. Heal their broken heart for his loss, in Jesus name I pray amen.