Cuche clinched the title in the Audi FIS World Cup Finals by five points over U.S. skier Bode Miller with fellow American Marco Sullivan in fourth.
The U.S. Ski Team said in a written statement that FIS rules require at least one training run before the finals, but soft snow on the Bormio course made that impossible.
Tuesday's developments left Miller in the lead in the overall men's downhill standings. He and Lindsey Vonn continue their quest to become the first Americans to win the men's and women's downhill titles in the same year since Phil Mahre and Tamara McKinney did it in 1983.
Vonn will be part of the women's downhill finals at Bormio that will take place as scheduled since the women were able to train Monday. The men have three more downhills on the schedule.