My friends dog was feeling threatened and nipped a family member's arm because they werent paying attention. The nip did no damage and was as far as I know, because he was scared. Most of the family wants to put the dog down because he's "dangerous" even though it's due to their own stupidity. He is a healthy 7 year old and it was my friend who raised him and is very distraught. The dog belongs to the family. A) can they legally put the dog down and B) could it be considered animal abuse?
As BlueDogRanch said, animals are property; you can have an animal put down because you consider it dangerous, because (if it is livestock) you wish to sell the carcass, or because you are tired of it. Legally, the dog belongs to the person whose name is on the dog licence; that person's view is decisive. (If you are asking whether the family should have a healthy dog put down, I doubt whether anybody on the internet can help; certainly Law.SE cannot).
And animal abuse would have to involve cruelty; use of the proper drugs by a vet definitely would not qualify.