When he treats Burnham and Raffi the same way, when he wants Seven marginalized, when he's harsher on Georgiou than he would be on Dukat, then it begins to paint a picture. When he thinks Jack should be made Captain in short order, it further supports that picture.
Then I lay out Burnham, Seven, Kelvin Kirk, and Jack Crusher's qualifications and the best he can come up with is, "But Jack Crusher has charm!" it reinforces that picture.
A misogynist is a misogynist, and I will point that out when I see it.
As far as "Nu-uh you don't like her cause you think girls are icky." No. More like, "He has the same mentality as those who used to say, "a woman's place is in the home!" We're only two generations removed from the world of Mad Men. The type of thinking they had back then didn't just disappear overnight. It just went underground and became more subtle. Sexism is still alive and well today, only camouflaged. Same goes for racism.