We introduce a new class of positive integers called Self-Healing Numbers, defined by a unique positional divisibility property. For each digit position i, the number obtained by removing the digit at position i must be divisible by i. We present complete computational enumerations up to seven digits, establish fundamental structural properties, prove non-hereditary behavior, and analyze the growth of the sequence. Our findings suggest a regular, exponential growth pattern, supporting the conjecture that this sequence is infinite. We also provide a more rigorous mathematical framework by expressing the divisibility conditions as a system of congruences, which allows for deeper theoretical analysis of the number's structure.