One way to approach this question scientifically would be to compare animals from one zoo to another. You could study overall health, reproduction, and lifespan of a species and see if there are correlations with enclosure size, diet, habitat enrichment, etc. This would give you information on whether specific zoo conditions are bad for the animal in question.
You could extend this by studying the same species in the wild, and collecting similar information there. If an animal usually lives longer in the wild than in zoos, for instance, then that is evidence that living in a zoo is not healthy for that species.
Another facet that should be considered in answering this question is whether keeping animals in zoos can help to prevent loss of genetic variability or even extinction. Many zoos now participate in scientific breeding programs designed to help protect vulnerable species from being lost.