In the realm of generalized modular metric spaces, we substantiate the validity of fixed-point theorems with Branciari contractions. This paper expands and broadens the original theorems in this context. Subsequently, by building upon this foundation, we explore various integral contractions to identify and characterize fixed points within this context. To highlight the practical implications of our work, we introduce the concept of the best proximity pair, thereby culminating in the best approximation theorem. We apply this theoretical construct to a specific example—one that is guided by the best approximation method described in prior research.