Yes it can. My mom constantly yelled at me during field hockey when I was younger. It makes us feel like we're not doing good enough, even when we try our best. It makes us feel like a failure and it really takes all the enjoyment out of sports. Also, he probably yells at her a lot at home too and it's probably been going on for a while. To a child, it's pretty much like mental abuse.
I can't stand parents like that. Thankfully, mine are proud of me no matter how well or how badly I do at a sport or whatever it is (grades are somewhat of a different story, but I still don't get yelled at until I cry and feel worthless). But I certainly think it can harm a kid emotionally, their self-esteem will be totally bruised. Man, if I was his kid I wouldn't stand for that ****. He might be the dad, but I'm thinking even I have a better head on my shoulders than him! Ha. I hope it doesn't continue (although by the sounds of it, seems like it will..), because some bad things could happen for her in the near future..
Mind your own damn business. He doesn't yell (embarrass) her in public. Since she is still skating I take it she has no problems with it. It might be the way they communicate. So but out, busy body.
I don't think parents should be coaches for their own children. As an answer above said, parents should be supporting their kids whether when they do well and when they do poorly. There are times when a coach can't offer such support because it would detriment the training. So when you combine the roles you get this, a child being scolded by their parent and feelings that they're not good enough to them fall into the mix.