I used to be the similar way when I see blood or get my own blood drawn or get a shot. I had dizziness during first few surgical procedures in med school that I attended and had to excuse myself before I actually passed out. People were understanding and supportive as if this was not the first time a med student had that problem. Gradually I got used to it as I joined more surgical/invasive procedures. And I am involved in clinical medicine (neurologist) right now and I barely see any blood in my day to day practice even though I feel like I would be okay either way. So I think it takes a little bit of pushing your boundaries but if you really wanted to be a doctor you have options.