I am having relationship issues!?

My domestic partner is sick with a rash. I got stressed out while cooking and started venting and asked if she would help (I may have said this with frustration). I continued that I am overwlemed and I am doing too much right now and wish she would help me because she usually helps out when I cook. She threw a fit and became physical with me and I yelled back at her and tried to prevent her from leaving the room. She said what I did was unforgivable and I shouldn't have gotten upset because she is not feeling well. She rolled her eyes and said how awful I am acting and I yelled at her that she is being unfair. She stormed upstairs and said what I did was terrible and I am a bad person. She also said she hated me and I yelled I hate her back. I went after her and apologized for what I did and told her that I take responsibility and I want to keep her happy. I explained that I was just stressed and hungry and that doesn't excuse my behavior but I would appreciate her understanding. She is extremely stubborn and I asked her to eat. She said she would only eat if I didnt talk to her for 3 days. I thought she was just saying this through hunger but it is not true, she says she is sticking to her words and said I am untrustworthy because I am going back on my word. After eating she said this is unforgivable and she will never forgive me. I told her I will not eat as long as she refuses to speak to me and I plan to stick with that. I have strong willpower I am able able to get through a long time with out eating. She said she would do the same and will refuse to eat until I eat. Is there anything else I can do to help with this mess. I love her and I dont want to lose her but I am sick of feeling this way and I'm tired of taking all the blame for every single fight we have. I dont know what to so I am completely broken hearted and I dont know how to deal with all these intense emotions. I cut myself to get some relief but it is not working. I am very depressed and I dont know what to do. I apologized and it doesn't work. I meant it and I apologised from the bottom of my heart. What am I supposed to do? I feel if we don't talk for 3 days it is proof that is relationship is over and there is no moving forward from thre. To me her refusing to speak to me is immature and a betrayal to me because she knows how it makes me feel. HELP!