If "sexist" means "believes that differences between the sexes make members of one sex competent at tasks for which the other sex is incompetent," then I'm a sexist.
If the implication is that anyone who acknowledges that the sexes are competent at different things is mean for doing so, then it's just a silly statement, and in a sense dishonest.
Whether or not they like to talk about it, there's no one who doesn't believe that the sexes are what you might call "differentially competent." Conservative or liberal, feminist or sane, you know that a man can't nurse a baby. Women can't play center in the NBA. Women are great at picking up conversational cues that men miss. Most people would rather have a female nurse take care of them in a hospital. These are unquestioned, even unconscious. Everything else is a negotiation over the specifics.
I say the job of Secretary of State is over the line. Apparently, you don't. Putting a Marxist-invented name on my person is uncharitable, even if vague. It also seems weak logically to make your case as, "Nice people don't question a woman's competence to do a man's job." It's using sensibility to avoid tangling with the practical issue—as if feelings were more important than reality. You can do better than that.
Was Margaret Thatcher a competent Prime Minister of Britain? I think she was, and was one of the best of the latter half of the 20th century.
You have the queer notion that women can't be diplomats or leaders of countries.
If it is not sexist, it is certainly luddite.