Cats are carnivores; vets never recommend vegetarian diets for cats. Vegetarian diets for cats exist, but they put the animal at risk for heart problems due to lack of taurine, and other conditions because they are missing lysine, vitamin a, and trytophan.
Dogs, who unlike cats are omnivores, can be fed a vegetarian diet, but they usually don't like it.
House pets, much like animals in the wild, need to eat the food that their bodies have evolved over millenia to eat. A person's moral objection to food should not prevent a beloved animal from eating the food that keeps it healthy. If an individuals personal beliefs make them willing to risk an animals health they need to re-examine what they believe. If a person objects to having meat in their home then get and herbovire, or just keep fish. I don't think it's animal cruelty, per se, but it isn't good, either.