I'm a US native speaker. Personally, I would be much more likely to say "I've never driven," but I think "I never drove" is correct, and common enough that I honestly don't even notice it and can't say exactly how common it is.
I think it's correct and quite frequently used.
It is very natural if it is said at the exact moment that someone is driving for the first time. Someone steps into a car and sits down at the wheel and says "Wish me luck! I'm excited! I never drove a car before."
It is also very natural if it is followed by words that explain "before what?" For example, "I never drove a car before automatic transmissions became common, so I never learned to drive a stick-shift."
Admittedly this is over a century ago, but Emily Dickinson wrote:
I never saw a moor,
I never saw the sea;
Yet know I how the heather looks,
And what a wave must be.