Purchase an ice-melting product that doesn't contain sodium chloride, or rock salt. As an example, IceClear is an organic product that safely melts ice without using sodium chloride.
Follow the manufacturer's recommendations when applying the ice-melting product. IceClear and other non-sodium chloride ice-melting products can be used to prevent ice from forming, so you can spread the product around your new driveway before the next big ice storm hits.
Use a bag of road sand to increase the traction on your driveway and prevent future ice buildups. Spread the sand liberally on your driveway.
Consider installing a heated driveway so you don't have to mess with ice-melting products in the future.