Moore, 25, is the first competitor to die in the 17-year history of the Winter X Games.
In a statement, family spokeswoman Chelsea Lawson said Moore -- a four-time X Games medalist -- "will be truly missed and never forgotten."
"The family wishes to express their deep gratitude for all the prayers and support they have received from all the fans, friends and family around the world that Caleb has inspired," the statement said. "They would also like to thank the physicians and medical staff at both Aspen Valley Hospital and St. Mary's Hospital, Grand Junction for their care and dedication."
The statement said the family would not make any other public comments or give interviews.
"We are deeply saddened by Caleb Moore's passing and our thoughts and prayers go out to his parents, Wade and Michelle, his brother, Colten, and the entire Moore family," said ESPN, which hosts the X Games.
"He will be remembered for his natural passion for life and his deep love for his family and friends, and he will always be an inspiration to everyone he touched in the action sports community," the cable sports channel said in a statement.
"As a result of this accident we will conduct a thorough review of this discipline and adopt any appropriate changes to future X Games," the statement said."