@OP your question is more getting at the roots of friendship and love. Take a look at how people become friends, and there's seemingly limitless ways, but if you start breaking them down, a common way male friendship works is through a shared hardship at some point in the past.
So often times if you look at a particular group of male high school friends, you'll find out they all had class with a particularly sexist history or math teacher at some point, and while they attribute their friendship to other things, that initial experience is what forced them to initially rely on each other, and that they could rely on each other gave them trust, and that is all that was needed.
There are men out there in the world I haven't spoken to in twenty years, but if you were in Mrs. Dunphy's class in seventh grade, you are welcome to sleep on my couch anytime, sort of thing. We didn't recognize that was why it was happening, and over the intervening years there were incompatibilities surfaced that fractured the group, but that core trust is still there.
So there's two things here, two ideas to think about:
1. Losing touch with people is a natural part of life.
Perhaps the least healthy thing about things like facebook is that it artificially prolongs relationships that had run their due course, and died of natural causes. You may be losing touch with this friend, and its up to you to decide if they are worth keeping in your life. It's up to you to decide if you can be adults that agree to disagree, if people are more than their beliefs. If you aren't particularly good friends, maybe its no great loss. But if they're a "help you move a body" type friend maybe you can tolerate a little bigotry.
2. Every argument is a choice you made. If the primary gripe is that you don't want to argue, stop arguing. Let them be wrong in their wrongness and get over thinking its your responsibility to engage.
The last bit I'll add is that my best friend and I agree on almost nothing. The one thing that keeps us going as friends is that we both have our shit together, which is a helluva lot more rare than either of us expected, so we can afford to meet and have a drink and not be doing something horribly irresponsible, and almost none of the other people we know have that luxury. I show up when he really needs help, and he shows up when I really need help. About the only thing we agree on is that we both really enjoy well crafted jokes about our own race. We disagree on religion, politics, some finer points of history. We enjoy discussions, not arguments. He and his wife will invite me over for dinner when there's been some particularly distressing spin put out by liberals on how conservatives have run amok, to hear my take on it, because I offer a perspective they don't get from their liberal friends. We don't agree, but they are reasonable thinking people, and we're able to lay out and dissect which points we don't agree. And for the most part, that willingness to really get down into the details, with that foundation of reliance and trust, such that the discussion doesn't turn to personal attacks questioning motivations for the argument, that's constructive.
I am better at everything else I do because of the interaction I have with them, and I have learned far more from them than from anyone who agrees with me.
Neither of us is particularly authoritarian though, so that helps. It might be different if they were more militant liberals, wanting to force their shit on me, or if I was a more militant conservative. My personal brand of Christianity includes a healthy dose of humility and non violence so I'm free to judge your wanton sinfulness but am not particularly required to make up laws about that shit.
Find common ground, focus on that. Avoid the mines, behave like adults when they come up. If either can't, especially if neither can't, better a former friend that an eventual prison sentence.