Honestly, once upon a time I would just keep arguing until it seemed completely futile.
But, some really nasty 'discussions' later, and just generally years (seriously. Me and forum arguments go back almost as far as the modern internet itself.) of being worn down by endless pointless arguments have led me to the point where I really just don't care anymore.
Often-times, if I get the impression there will be something really unpleasant involved, I will literally not even read the replies I get. I just don't want to deal with that stuff anymore.
If that comes across as rude? Fine.
If it means I end up ignoring people that actually made perfectly reasonable and civil replies to what I said?
Also fine.
I just don't want to deal with the kind of sickening crap I have waded through in the past.
I'm done with it. Can't really cope anymore.
The only obvious next step is to simply not get involved in the first place.
Which... About half of the time is indeed what I do as well.
Feel the urge to reply to something? ... I end up looking at whether I really want to deal with the possible conversation that might result, and then... Often choose not to get involved.
Sometimes, my... Irritation with something gets the better of me and I post something anyway...
More often than not I end up regretting it. (which is how I end up getting a bunch of replies I never read)
Forum replies go unread by me.
Youtube comment notifications? Definitely ignore those.
Facebook? Eh. Hard to avoid.
Other things? Yeah... Ignore those too a lot of the time.
I just can't really handle it anymore. So... I don't.
But in general there really isn't a 'polite' way to end an internet discussion anyway, so worrying about whether you're being rude or not is kinda pointless, because realistically, you're going to come across as somewhat rude in this kind of situation no matter what you do...