Can a driver race his own car in Nascar?

You CAN, but most people don't. Tony Stewart is an owner-driver; he owns the cars he races. Dale Earnhardt was an owner-driver too, but he owned cars that other people drove, and drove cars other people owned. Darrell Waltrip and Bill Elliott were also owner-drivers, as was Richard Petty.

If by "his own car" you mean the one in his driveway...no, street cars are very different from racing cars.