This situation happened today: A dog was attacked by another dog on shabbos and was injured. The owner of the injured dog, a Jew, took the dog to the animal hospital via automobile, because he did not want the animal to suffer anymore. Was this halachically permissible?
No. It was not the correct thing for him to do. We are only permitted to be mechalal Shabbos in order to save another Jew’s life, and in most instances even the life of a non Jew. Although we have a mitzva not to cause an animal suffering, it is not one of the things for which we may be machalel Shabbos. In fact even a melacha d’rabonon is also not permitted. It would however be permitted for a gentile to do it for him, in order to prevent pain to the animal.
MIshna Berura 332-6.