YES! Actually you can heal incipient caries (small early lesions in the surface of the enamel), without having a dental procedure completed. A cavity begins when the acids released by the bacteria on your teeth begin to decalcify the minerals of your tooth enamel. As this process continues and the bacteria continue carbohydrate metabolism, a small hole begins to form in the enamel, from the acid induced decay. When this hole has just begun to form and is very shallow, it becomes visible on a dental radiograph (x-ray). This is the stage when it is still able to heal naturally. But, this requires you to make some changes in your habits. Brushing AND flossing every night, using a fluoride mouthrinse, and decreasing the frequency of carbohydrates in you diet are the best way to reverse this process. Your dentist may also prescribe a high strength fluoride toothpaste to aid the repair process. When you do these things, two important things happen. First, you remove the bacterial colonies from your teeth so they cannot continue releasing the damaging acid that causes decay to progress. Second, the tooth enamel can begin a process of repair by recalcification of the damaged mineral structure. The minerals present in your saliva aid in this process, as well as the fluoride in your toothpaste, water and mouth rinse. In fact, when the fluoride ion combines with the minerals in tooth enamel, it actually becomes stronger and less susceptible to decay. Now, if the process is not reversed in this initial stage, the cavity will begin to grow bigger and further into the tooth and will not be able to naturally heal. Once it progresses more than halfway through the enamel, the cavity needs to be repaired with a dental restoration (filling). If not, it will progress to the dentin (the softer inner structure of you tooth), where it will spread and eventually reach the pulp (the area where the nerves and blood vessels are).
So in addition to good home habits, the key to avoiding the need for dental procedures is having regular exams, so any issues can be caught early and treated in the most conservative manor.