I don't think you are being unreasonable to be concerned, but I do think you have gone about things in entirely the wrong way!!! I totally understand you checking her emails and I will do the same with my kids at that age, but to tell her that's how you found out?? Big mistake, huge classic rookie mistake - what we're you thinking?? Of course she was going to freak that you invaded her privacy! And on top of that will feel humiliated that you read her private thoughts, her confidence will have taken a knock from that pushing her to cut more if that's what she's doing, and to clam up even more so you'll get even less out of her! You said she's a bit chubby, well all 15 year old girls have puppy fat, but if she is overweight and you found out she felt like that you should have gone on a mission to change all your lifestyles together in a healthy way, join a gym together or regular swimming or running, do it as a get fit thing rather than a lose weight thing but at least then you can watch her and control it more. I'm not sure what you expect a GP to do - pump her full of anti depressants as a quick fix or stick her on the year long NHS waiting list for 6 sessions of counselling??? You confronted her, it was an ambush, an attack, she's not going to be more willing to come forward and talk after that! You need to apologise to her, properly, heartfelt and with no agenda. No sorry BUT I was worried. Just sorry, admit you were wrong, promise to never do it again (and get caught!) and acknowledge how it made her feel. Then you have to work on getting her trust back, not obviously, but take her on a mother daughter spa treat or something, or even just a lunch. Hopefully, in time, she'll start talking to you. Maybe grotty kids at school have said something, maybe a boy she likes is dating a skinny girl, the F U attitude is always just a cover and always from the most insecure people deep down. Re the cutting I would try and catch her out "accidentally" when she's changing or something, ask the PE teacher at school if they have seen anything, the spa day or swimming would force her to show some flesh in front of you. For the record I have been shaving my legs for 23 years and not once managed to do it without cutting myself somehow! She could genuinely be worried about her thighs, but even if it is cutting don't panic! It's not nice for your child to be doing that to themselves but as a previous cutter it does serve a purpose and does not automatically mean they are going to progress to slashing wrists! Cutting is a way to release pain, when emotional turmoil is so extreme you can't cope anymore a cut focuses you, stops the pain, calms everything and gets you back on track. She needs to learn other ways to deal with panic and hysteria, try mindfulness, meditation, yoga, again a spa day might have introductions to this and it could be something you do together. If she is cutting obviously she needs to sort out what is causing that pain, and private counselling, though expensive, would be best for that. But it has to be at her pace with her agreement in a way that supports her and builds her confidence, you can't force her with this, she'll just dig her heels in more and be more resilient. Please don't panic, it could be something, it could be nothing, but I think you need to calm it down and back off and bit initially, slowly slowly catchy monkey. Let her feel safe and in control, let her come to you (or your DH or a favourite aunt or grandmother) and then face things with her together. I never cared about my weight really but I did have an "I'm so fat" phase to keep up with every single other girl in the class who was saying that. 
Ludicrous really.