Some people maintain that it is only obligatory to feed pets but not to treat them when they are ill. What is the correct opinion on this matter?

Raising animals at home necessitates providing them with good treatment and care. A raised animal is entitled to regular and suitable food and drink that meet its needs, medical treatment, and safe shelter.
These are the simplest rights of animals guaranteed by all faiths.

Islam enjoins mercy and kind treatment to all creatures. The Prophet (peace and blessings be upon him) informed us of how to treat animals. He mentioned that a man entered paradise for quenching a dog's thirst. He also informed us of a woman who entered hellfire because of a cat which she had tied. She did not give it food to eat nor set it free to eat from the vermin of the earth.

Islam goes even further than showing kindness to pets we raise at home. The Prophet (peace and blessings be upon him) commanded us to be kind to animals allowed for consumption at the time of their slaughter. He taught us not to allow the animal to see other animals being slaughtered or to show it the knife used. In addition, the knife should be sharp enough to facilitate the slaughtering without causing pain to the animal.

And God Almighty knows best.