No, rabbits are not originally native to America. The species of rabbits we commonly see today, such as the Eastern Cottontail, are native to North America, but they evolved after the arrival of other rabbit species from Europe.
The domesticated rabbit that we often keep as a pet, the European Rabbit (Oryctolagus cuniculus), was introduced to America by European settlers in the 17th century. It has since established wild populations in various regions across the continent.
In contrast, the rabbit species that are native to North America include various cottontail species like the Eastern Cottontail (Sylvilagus floridanus) and other native hares. These native species have adapted over time to their environments, and they differ from the European rabbits that were introduced.
In conclusion, while some species of rabbits are native to North America, the European rabbit most people think of when considering domesticated rabbits is not native to the continent.