We know that we are not to treat animals cruelly and that we are allowed to kill an animal if it is a threat to our life … but is it okay to hunt animals for sport?
God entrusted animals to the stewardship of those whom he created in his own image. Hence it is legitimate to use animals for food and clothing. They may be domesticated to help man in his work and leisure. Medical and scientific experimentation on animals is a morally acceptable practice if it remains within reasonable limits and contributes to caring for or saving human lives (CCC 2417).
It is contrary to human dignity to cause animals to suffer or die needlessly. It is likewise unworthy to spend money on them that should as a priority go to the relief of human misery. One can love animals; one should not direct to them the affection due only to persons (CCC 2418).
God gave man stewardship over animals, and that includes using them for just purposes. Examples of just purposes that the Catechism points out are food, clothing, medical and scientific experimentation, and the work and leisure of man. The only caveat is that man must not cause animals to suffer and to die needlessly because to do so would be contrary to man’s own dignity. Animals, properly speaking, do not have “rights” because they are not human. But man does have the human responsibility to treat them with reasonable care.
Does this exclude sport hunting? If a hunter were merely shooting an animal for the “pleasure” of watching it suffer and die, yes. But the vast majority of hunters don’t do that. Some use the meat and skins of the animals. Others are helping to preserve the balance of nature by using carefully-regulated licensing procedures to thin out animal overpopulation. All responsible hunters take care not to leave a wounded animal, injured by a badly-aimed shot, to suffer but make sure to try to track it down and end its suffering.
In short, the Church does not oppose sport hunting.