Before you lay landscaping fabric under a gravel driveway, there are several things that you need to consider. In most cases, traditional landscaping fabric will not be strong enough to hold up to the wear and tear of cars driving on it repeatedly. It might last for a few years, but it will start to tear apart soon.
One option that you have is to use geotextile fabric instead. This is a little stronger than traditional landscaping fabric. The easiest solution may be to purchase a commercial herbicide and spray the driveway every year.