I don't believe the alkaline nature of the salt is doing any "damage" at all to your floor. It is simply residue. To remove the residue use an acid based cleaner like This Old Grout Easy Etch, Viper Renew, Prochem makes one also. These cleaners are buffered as to not etch the grout too much. The trick is to extract the water as opposed to mopping it up. Use a vet vac as mentioned above. With a mop you are picking some up while just redepositing the rest and when it dries, you have residue. After tou are done, go back over with an alkaline cleaner to neutralize any acid cleaner left.