Sexism refers to a belief that one particular group of humans are inherently inferior because of thier sex, and that they ought to be treated as such. Is this a fair definition?
If so, I think that we can be fairly certain that the Catholic Church strongly adheres to a sexist conception of reality.
It isn't so surprising of course. The early Christians were far more progressive than most of thier contemporaries, but growing in the context of Mediterranean societies that considered free men to be everything and all othjers to be little better than property (children, women, slaves) it is not so surprising that all major Christians internalized some sort of conception about the inferiority of women.
Though the bible never explicitly states "Women are worse than men. Full stop" it goes on huge ramblings about the uncleanliness of women, about the irrationality of women, and about how women often have problems guiding themselves. Women, it is often stated, must subject themselves to the dominion of man in order to succeed. Classic patriarchial assumptions.
As the bible oft repeats,
"Let wives also be subject in everything to their husbands... For the husband is the head of the wife as Christ is the head of the church" (Ephesus. 5:24)
"I suffer not a woman to teach, nor to usurp authority over the man, but to be in silence." (Timothy 2:9)
Regarding uncleanliness
"If a woman have conceived seed, and born a man child: then she shall be unclean seven days; But if she bear a maid child, then she shall be unclean two weeks" (Leviticus 12:5)
"How then can man be justified with God? or , how can he be clean [that is] born of a woman?" (Job 25:4)
This implies, even if it does not say explicitly, that at least in terms of agency, women are inferior to men. That women are assumed to be physically inferior is a given. Personally, I think that agency is the most important quality of humans, so this is pretty major.
Thomas Aquinas, too, believed in the subjagation and inferiority of women, despite believing that they too deserved some of the rights of men.
"The male sex is more noble than the female, and for this reason he [Jesus] took human nature in the male sex" (Summa Theologiae III:31:4 ad 1).
More recently, the early eucuminical councils decided to bar women from being part of any Church authority structure. The example of Jesus Christ was forgotten or ignored. Given the rampant sexism in both society at the time and many religious texts, it is totally unsurprising.
Currently, the Church defends it's practice of barring women from preaching by cllaiming that it is impossible to alter church teachings. It's really to bad that the argument is undermined by centuries of altering the catechism, changing and adapting to meet a changing world.
Nowadays, the church also stringently opposes abortion, but not only that, it considers contraception a sin as well, equal in gravity to abortion. Not only is equating the two ludicrous, but it is important to note that an essential element of subjecting women to the dominion of a man is to get her pregnant against her will. To escape that necessity threatens the dependency of women as a lesser class of human.
Finally, Pope after Pope has denounced feminism, it has denounced women raising children with no husband, it has denounced all manner of actions desinged to give women indepedepence from men. The reasons given are same one's given by any other partriarchy. Stability, the rationality of man, the rights of the child, the inferiority of women in a physical sense....
The Catholic Church is a sexist organization.