Intro: My good friend talked major shit behind my back.

Context: I'm a student and rent a house with 4 other dudes, one of them is my gym buddy (we go to the gym 4-6 times a week since the school year started), share food together, cook together, shop together, go out etc. So, basically we got really close but not best friend close as we have incredibly different world views, mine are quite conservative, TRP-based views, his are SJW, totally blue views.

One of my best qualities is being loyal to my friends, always sticking up for them and never talking shit behind their back (outside of our house).

Today, a mutual female friends of ours who literally records her whole lifestory on snapchat posted a snapchat of my friend saying "Yeah he definitely has some issues - sexist, racist, homophobic, doesn't like his hair being touched". They must have been talking about me and this was obviously about me in front of our mutual social circle (which includes fit girls that are obviously on my radar).

My friends who looks like a total alpha, is a total beta nice guy who believes in equality, feminism, politically correct EVERYTHING and obviously disagrees when I say that men and women and totally different so I would not label them as equal, or when I said that I would find it weird if gay people were suddenly allowed to get married etc. (Obviously this was all said when all 5 of us are in the house chilling together and I would never share it with anybody outside of the house, unlike him apparently...)

This all happened a few hours ago, what should I do about this behaviour? I will definitely speak to him face to face to explain the whole thing but I am not sure how to deal with our friendship in the future. Should I demote him to just a housemate and stop going to the gym and cook with him if this is how he treated me? I know this is long but I really want to hear from all of you!