Completely ignoring the present argument and going back to the original conversation... really easily, actually.
Step one involves everyone pausing in sneering at each other. Gamers, among other things, want really good games to play. Social justice warriors want a world that stops shitting on people simply because they happen to not be cis heterosexual white dudes.
You know one of the ways to get really good games to play? Tackling interesting real-life issues and characters beyond Generic Grizzled White Marine Dude No. 1247. Taking real-life struggles into account. Realizing the depth of characterization you can get if you go beyond cookie-cutter hero types and accepting that our women characters in video games must be eye candy, our gay characters in video games must be jokes, our black characters in video games must be prime sidekick material ready to be killed for a cheap shot of 'emotional distress' even if we've been given no reason to care about any of the characters, et cetera, et cetera.
The most important part of that step one is realizing that both sides want to make the world better. Gamers want to very specifically make the world better by encouraging better games to play. Social Justice folk are also wanting to make the world better because they experience (or see their friends experience) all of this additional bullshit that doesn't even make sense being piled on top of them for arbitrary reasons.
Step two is putting this into practice. Recognize that yeah, there are some SJWs that go too far and get too mean, just like there are some on the other side who do the same thing (sup 4chan users who plan to save selfies posted on tumblr and photoshop them onto porn and spread them about the person's irl contacts for the express purpose of ruining their lives).
Come to the table not as adversaries but as allies. Stop asking "why are you so wrong", but "how can we work together". Listen to one another.
Realize there's going to be anger on both sides. For social justice folk especially. I'm just your run of the mill straight WASP gal, but when I see the kind of stuff some of my friends go through for being, for instance, asexual, it makes my blood boil. Who in their right mind thinks it appropriate to threaten to rape somebody to 'cure' their sexuality? Who in their right mind thinks it's okay to send death threats and cut off all ties with somebody simply because they're not interested in sex, for whatever reason? And that's the tip of the iceberg. Once you see what kind of things are going on to you, to your friends, it's really easy to get angry about it. Because it's worth getting angry about!
Imagine all that anger, and the feeling of noticing it every single day and being overwhelmed by it. That's why social justice people get so angry and sometimes get defensive and overboard. I know it might seem silly, but it's a case of sitting down and having a little empathy. Even if you don't understand every detail, take some deep breaths, and realize that people are hurting and being hurt.
I realize it's kinda difficult sometimes. But the two groups are actually natural allies. I mean, gamers want better games, and part of better games is making better characters. One of the many things social justice folk want is better characters in media to represent everybody fairly instead of just generic white dudes.
It just takes stepping back and not thinking of each other as adversaries, but as allies, and calmly and respectfully listening to one another.
...yeah I know that's pretty impossible, but hey, worth a try, right?