It looks like you have a lot of thoughtful responses, but I feel compelled to echo others statements because this exact thing happened to me when I was in high school.
For me, I think it was a combination of shyness/social anxiety and stressful home life compounded by the "not fitting in" that every teenager goes through. I would describe it kind of as burn-out. Unfortunately for me, part of my stressful home life meant that my parents weren't very diligent about pursuing options for me, although they did take me to the doctor for blood tests, and put me on an anti-depressant, but like your daughter, I refused to see a therapist. I ended up basically dropping out for the second semester of my junior year. Fortunately for me, I did continue going to my ballet classes in another town, which was a bit of a haven for me (I never had any anxiety about going there) and because I was taking advanced classes and had a lot of credits already I actually ended up graduating ahead of schedule through a combination of independent study and taking one semester of classes my senior year. Also, I should mention that I decided on my own to go off the medication after about 6 months and have never been on any sort of medicine since. I don't think it was necessary for me. (However, this was long enough ago that there were only 2 anti-depressants on the market.)
Of course, I can't know how things would have turned out if I had done things differently, but at the time I wanted to apply to see if I could just go right to college and my HS counselor talked my parents out of it. I think it is good to listen to what your daughter is telling you, and make sure you do research on all your options, homeschooling, online academies, other local schools, work-study programs, boarding schools, etc. (Check out www.scattergood.org, my brother went there). Perhaps you could tell her that she needs to go to school while you research your options, but that you want to work together to figure out a way to make her happy.
It might be helpful to ask her questions like, "In what situation would you feel comfortable going to your school?" or "Who would you feel comfortable talking to?" and helping her find places that she isn't so anxious and building from there. I don't think it's fair to expect kids to talk to a school counselor, so maybe you can research together to find a therapist that she would like. Again, there are lots of options, check out http://www.arttherapy.org/ My best friend is an art therapist and I think it's especially effective with children and adolescents.
I would be very concerned about homeschooling, due to the social anxiety issues, as others have mentioned. I have found that, for me, the more I am around people and in social situations, the fewer issues I have, but it is still something that I have to work on. You also need to consider how self-motivated she is and both your and her personalities and be realistic about whether it is a good solution for you.
I also want to give you hope that she (and you) will find a way to get through this, and she can go on to be very successful and happy in her life. I really feel for you. Please let me know if I can help.