Kathryn,you can bring any animal to the vet and have them put down,healthy or not,young or old and people do,once they don't want their animal anymore
,there is no law saying you cannot.
Pets are considered property and can be discarded just like so much garbage.
Often time a compassionate vet will give the animal a chance,but needs permission from the owner,and try to find a home for the doomed cat/dog.