No, domestic cats are not native to North America. In other words, the domestic cat does not originate from a wild cat species in North America. You can tell that because the only small wildcat that could possibly have been the precursor to the domestic cat is the ocelot which at one time lived in North America but no longer (although some people claim it still exists in America). The domestic cat could not have originated from the ocelot because it's too large. The domestic cat is a domesticated and socialised North African wildcat. So the domestic cat is native to Africa and the Middle East. The domestic cat exists in North America because they were imported by Europeans to the east coast in the 1600s.
The cat hating fraternity like to remind the cat loving fraternity that the domestic cat is an invasive species in North America. And because the cat is an invasive species and because they prey on birds there are some people who think that the domestic cat should be exterminated in North America. These are ornithologists and bird lovers! The problem with the idea that the domestic cat is non-native and an invasive species is; when do you start considering a cat to be part of the country? If a domestic cat has been in the country for 400 years I think you can say that the cat is no longer non-native. Well technically you can't say that but over such a long period of time the concept of "non-native" becomes a bit pointless.