So many things to say about this!
First of all, congrats to Naomi - I remember being impressed with her at Wimbledon a few years ago, I couldn't believe how young she was, and I think she'll be one to watch over the next few years.
Secondly, the crowd behaved appallingly, but that is par for the course for the US Open crowd - they were also awful during Madison Keys quarter final!
The coaching rule is stupid imo. Everyone is doing it, and it just seems your luck whether the umpire decides to call them out on it or not. But absolutely every coach does it. Not sure if it is fair to penalise a player for the coach's actions. Perhaps it would be better to banish the coach from the stands?
Imo, the umpire did not do his job properly and he inflamed a tense situation. He should have given Serena a warning, that if she didn't calm down then the next step would be a game deduction. He is an experienced umpire, so I don't know why he didn't use his discretion here, other than the fact he was on a power trip.
Serena was obviously stressed, and I think if it had been anything else that she got the initial warning for she would have let it go, but she was furious that the umpire thought that she was getting coaching. I can understand how angry and upset you can get when someone believes something about you that you know is wrong.
I've been thinking about this all night, and absolutely this was sexism. Male players would never be treated as harshly. Serena has a valid point. Not excusing her behaviour, but I can understand how in the heat of the moment she was upset, and again, this is where a good umpire would have calmed things down and allowed the match to be completed properly. Serena wasn't going to win this anyway, and his actions robbed Naomi of a "proper" win.
Serena has had to fight inherent racism and sexism throughout her career, so I do not blame her for her outburst. It must be so tiring to have to fight it all the time. And there are prime examples of the attitude towards her and her sister here on this thread!