Cormier, 57, ascended the south side of the mountain with a group of climbers, and at one point complained of a leg cramp.
He reached the summit alone, and at about 8 a.m. fell from a 1,000-foot drop while looking over the east side of the mountain.
Air rescuers photographed Cormier, who was caught on a rock ledge, but he appeared to be lifeless, the Hood River County Sheriff's Office said.
"It was a true shock as we learned today about his passing, and we're certainly very concerned about his family, concerned about his parish, and all the people who've been part of his life," said Jim Goodness, spokesman for the Archdiocese of Newark.
Rescuers said they would attempt to recover Cormier's body next week, once colder temperatures allowed more stability on the mountain.