A short story, probably from the 60s, where rather than go through the mess of fighting an interstellar war, champions fought in an arena. I think these battles between champions happened regularly. I know this plot is almost a cliche, but what made it memorable to me was at the end the earth guy wins and he comes home disillusioned and says he could do anything he wants and no one would care, so he goes into a house and murders a young girl.
Sounds a lot like "The Survivor", a 1965 novelette by Walter F. Moudy, also the answer to this old question and this one; first published in Amazing Stories, May 1965, available at the Internet Archive. You can also read it here.
"The Survivor" differs from your description in that there are no interstellar wars; the fighting takes place on Earth, between Americans and Russians. The battles are not between single champions, but between armies of 100 men from each side, fighting in an arena until one side is wiped out. The survivor is immune to all laws, and at the end of the story the survivor (an American) walks into a neighbor's house and goes into the young girl's bedroom. We do not see what he does with the girl. I assumed that he just raped her; it didn't occur to me that he could have murdered her as well. The ending is already spoiled but I'll spoiler-protect it anyway:
The Martins were still up. Mr. Martin was tying brightly colored flies for his new fly rod and Mrs. Martin was reading. They both stiffened when he entered without knocking —alarm playing over their faces like a flickering fire light. He didn’t pause, but walked on up stairs without looking at them.
Mrs. Martin got to her feet and stood looking up the stairway without moving. In her eyes there was the look of a jungle tiger who watches its mate pinned to a stake at the bottom of the pit. Mr. Martin sat staring at the brightly colored flies on his lap. For a moment there was silence. Then a girl’s shrill screams announced to the Martins that war’s reality was also for the very young.
The classic story about an interstellar war decided by single combat between a representative of each side is "Arena" by Fredric Brown (first published in Astounding Science Fiction, June 1944, available at the Internet Archive). In Brown's story the battle in the "arena" is not a regular way of settling interstellar conflicts, it only happens once. The soldier from Earth wins, but nobody else knows about it, so he does not get any recognition for saving humanity from destruction, much less a license to rob, rape, and murder.