You'd ask yourself just how diverse the wildlife in an urban environment is, where the bulk of domestic cats live. Right now I'm not convinced about the ecological impact.
Ironically, they're in the decline in their natural habitat while thriving in North/South America. The Americas has a few native pigeons, but none are as well known or wide spread as the Rock Pigeon. Quite a few pigeon species are also far from the ugly "rats with wing" stereotype[1], but they don't live in Europe or North America.