I am not exactly sure what classifies .... A Climbing Gym. There have been many climbing structures built pre 1980s for climbing schools, many boy Scout camps had a climbing structure, some weight lifting gyms glued rocks to their concrete walls and called it a climbing challenge, etc, etc. The Vertical Club in Seattle Washington is listed as being the first Climbing Gym in the USA, so I will start there.....
Vertical Club, opened fall 1987 (listed as being the first Rock Gym opened in the USA) Owners: Richard Johnston and Dan Cauthorn Seattle, Washington
Vertical Club - January 1988 advertisement
Vertical Club - August 1988 advertisement
Vertical Club Richard Johnston and Dan Cauthorn
Portland Rock Gym - opened early 1988 Portland, Oregon
SCS - Portland Rock Gym - March 1988 advertisement
Portland Rock Gym - September 1989 advertisement
Cat's Gymnastics Boulder, Colorado Opened summer/fall 1988
Rob Candelaria
Boston Rock Gym - Opened August 1989 Boston, Massechuttes
Boston Rock Gym July 1989 advertisement
Art Of Climbing Opened August 1989 Chattsworth, California
Art of Climbing - August 1989 advertisement
Extreme Sports - Opened February 1990 Grand Rapids, Michigan
Extreme Sports - February 1990 advertisement
The Sporting Club - Opened June(?) 1990 Atlanta, Georgia
Entre Prises advertisement
Entre Prises - June 1990 advertisement
Colorado Indoor Climbing Center "Coming Soon" - June 1990 ........., Colorado
Colorado Indoor Climbing Center June 1990 advertisement
The Wall Climbing Center "Coming soon" - June 1990 Brooklyn, New York