Purpose: The importance of robust proton treatment planning to mitigate the impact of uncertainty is well understood. However, its computational cost grows with the number of uncertainty scenarios, prolonging the treatment planning process. We developed a fast and scalable distributed optimization platform that parallelizes this computation over the scenarios. Methods: We modeled the robust proton treatment planning problem as a weighted least-squares problem. To solve it, we employed an optimization technique called the Alternating Direction Method of Multipliers with Barzilai-Borwein step size (ADMM-BB). We reformulated the problem in such a way as to split the main problem into smaller subproblems, one for each proton therapy uncertainty scenario. The subproblems can be solved in parallel, allowing the computational load to be distributed across multiple processors (e.g., CPU threads/cores). We evaluated ADMM-BB on four head-and-neck proton therapy patients, each with 13 scenarios accounting for 3 mm setup and 3:5% range uncertainties. We then compared the performance of ADMM-BB with projected gradient descent (PGD) applied to the same problem. Results: For each patient, ADMM-BB generated a robust proton treatment plan that satisfied all clinical criteria with comparable or better dosimetric quality than the plan generated by PGD. However, ADMM-BB's total runtime averaged about 6 to 7 times faster. This speedup increased with the number of scenarios. Conclusion: ADMM-BB is a powerful distributed optimization method that leverages parallel processing platforms, such as multi-core CPUs, GPUs, and cloud servers, to accelerate the computationally intensive work of robust proton treatment planning. This results in 1) a shorter treatment planning process and 2) the ability to consider more uncertainty scenarios, which improves plan quality.