This study introduces the Iterative Chainlet Partitioning (ICP) algorithm and its neural acceleration for solving the Traveling Salesman Problem with Drone (TSP-D). The proposed ICP algorithm decomposes a TSP-D solution into smaller segments called chainlets, each optimized individually by a dynamic programming subroutine. The chainlet with the highest improvement is updated, and the procedure is repeated until no further improvement is possible. We show that the subroutine runs in quadratic time and the number of subroutine calls is bounded linearly in problem size for the first iteration and remains constant in subsequent iterations, ensuring algorithmic scalability. Empirical results show that ICP outperforms existing algorithms in both solution quality and computational time. Tested over 1,249 benchmark instances, ICP yields an average improvement of 2.6\% in solution quality over the previous state-of-the-art algorithm while reducing computational time by 91.3\%. The procedure is deterministic, ensuring reliability without requiring multiple runs. The subroutine is the computational bottleneck in the already efficient ICP algorithm. To reduce the necessity of subroutine calls, we integrate a graph neural network (GNN) to predict incremental improvements. We demonstrate that the resulting Neuro ICP (NICP) achieves substantial acceleration while maintaining solution quality. Compared to ICP, NICP reduces the total computational time by 28.6\%, while the objective function value increase is limited to 0.14\%. A transfer learning framework enables efficient extension to various operational constraints, making this a valuable foundation for developing efficient algorithms for truck-drone synchronized routing problems.