Be sure to scroll down... there may be more than one question on this page!
First things first… you have to kill the existing weeds. Boiling water is a tried and true natural remedy for weeds, plus it will also kill the seeds! Of course, for larger areas boiling water is impractical. The USDA Agricultural Research Service has found that spraying vinegar on weeds will effectively kill them when sprayed on the leaves. They found that typical 5 % (acidity) white vinegar easily kills young weeds, while more concentrated vinegar (over 10%) will kill adult weeds. More info at this link: http://www.ars.usda.gov/is/pr/2002/020515.htm
Now, depending on your "green" temperament, you can either maintain the driveway with the above techniques or use a weed "blocking" chemical to prevent regrowth. You need not hunt for an exotic product. One intriguing natural solution to weed growth is corn gluten meal. This natural byproduct of cornstarch manufacture works by stopping the growth of "feeder roots" in seedlings but does not affect other soil organisms, bugs or growing established plants. Since it breaks down in the soil, it does not cause any runoff issues for lakes or streams and is not hazardous to animals or people.
Of course, the same chemicals that are used on lawns to block weed growth can also be used on a stone walkway, such as Preen. These products utilize chemical growth inhibitors which, though not shown dangerous to animals except in very large exposures, are nevertheless are not "natural" products.