C.M.
We had a lot of resistance from family members when we started 3 years ago. My mom and dad thought we were nuts, and other family members made comments here and there. It still happens, we get a snarky comment here or there and yes, people do blame homeschool for every little thing that goes wrong!
Instead of changing people's minds (which is hard to do and quite pointless) I've learned to get support from other homeschooling families in my area. When things get hard, I talk to them. I can't talk to my mom, because she just says "well, if you'd just send her to Public School you wouldn't have that problem." It's such a relief to talk with other people going through the same journey you are.
I've let the proof be in my daughter and her progress. I don't discuss homeschooling with my family and if it comes up I change the subject. It's not worth discussing, they're not changing my mind and I'm certainly not changing theirs!
Since we've started 3 years ago we've had some very positive remarks from family members who were naysayers in the beginning. One commented on how grown-up she handles herself. Another was surprised when she jumped in and joined a science discussion with my cousin who is 3 years older than her! She knew more than him about astronomy!
My family does remark about all the wonderful field trips we take and how lucky she is to be able to experience things first hand.
If I were you I'd take the "it's your child, do what you want" and be okay with that for now. At least they aren't nagging at you every minute! The proof will be in what a wonderful and well-educated lady your daughter will turn out to be. As she grows they will see--and that is all the argument you need.
Good luck!