He has no pity for anybody.
The double negative in the German sentence bugs me. It’s literally “he knows no pity towards no one”, which sounds wrong to my English-speaking brain. I would have expected “er kennt kein Erbarmen, gegenüber jedem” but that’s probably wrong.
This is not a double negative, the comma is the clue here.
Without the comma it would be wrong, but here “gegenüber niemandem” is kind of an afterthought or more likely a confirmation.
You could also use a dash or even a full stop.
A better translation:
He knows (has) no pity. For no one.
Your proposal “Er kennt kein Erbarmen gegenüber jemanden” is not correct.
If you want to put it in one sentence:
“Er kennt für niemanden Erbarmen.” but that is less stong than the one above.
That would still be an incorrect double negative in English. If you want to end the sentence dramatically with “for no one”, then the “he has” part must be expressed in the positive:
“He has pity” (positive) + “for no one” (negative) = negative meaning, he is merciless
“He has no pity” (negative) + “for no one” (negative) = positive meaning, he has pity for someone
Put it all together and you would end up with:
He has pity for no one… (dramatic pause)… for no one.
I like your “less strong” sentence in German better, because it is closer to how I would say it in English.
He knows (has) no pity. For no one.
It’s the old problem of grammar versus usage, isn’t it? Issues like double negatives are only queries for people like “us”, you know, the ones hanging out in language forums. I can’t imagine anyone else being confused by this statement. Not anyone. 
Oh, absolutely - it’s a pretty mild mistake as far as mistakes go. The only places I would ever get pedantic about double negatives are on language learning forums like this one, or on my YouTube channel: LGFTV013 - We don't need no double negatives! - YouTube 