We present a data-parallel algorithm for the construction of Delaunay triangulations on the sphere. Our method combines a variant of the classical Bowyer-Watson point insertion algorithm with the recently published parallelization technique by Jacobsen et al. It resolves a breakdown situation of the latter approach and is suitable for practical implementation because of its compact formulation. Some complementary aspects are discussed such as the parallel workload and floating-point arithmetics. In a second step, the generated triangulations are reordered by a stripification algorithm. This improves cache performance and significantly reduces data-read operations and indirect addressing in multi-threaded stencil loops. This paper is an extended version of our Parallel Processing and Applied Mathematics conference contribution. Copyright (c) 2016 John Wiley & Sons, Ltd.