Several people have suggested groups such as NOLS and Outward Bound. Both are very worthy groups, and I will chime in with another group, albeit with a slightly different mission.
One area that fueled my sister's interest in the outdoors as a teen (to her credit, she got me involved in backpacking many years ago) was a summer stint with the Student Conservation Association (www.thesca.org)
As part of a volunteer National Conservation Crew, she traveled to Idaho to build trails for six weeks with other 15- to 19-year-olds lead by one or more crew leaders.
The SCA is an organization that's been around for more than 50 years, sending teens into national parks, wilderness, and other federally protected lands doing a bunch of the work that we, as backpackers, often enjoy.
The cost is pretty nominal - there is a small application fee and on top of that, a plane ticket to wherever she is sent (they generally try to get kids to go to a different part of the country). The application process makes allowances by the experience of the applicant - in my sister's case, she had back country experience and they sent her deeper into the woods. My friend’s son, on the other hand, had little experience in the woods and was sent to work on trails at Cumberland Gap National Historic Park.
I don't know if you can make a kid enjoy the outdoors. But I do know my sister still speaks glowingly of the SCA. I ran across a group from the SCA last summer building trails in the Desolation Wilderness (pretty good gig, huh?) and I got a chance to visit with them for a while and thank them for their work. Most of the kids seemed to enjoy the opportunity to spend their summer with others their age. Most were enthusiastic about the woods and camping. The summer of volunteer work, while demanding, gave them a great sense of accomplishment while introducing them to the outdoors.