Which fight would have changed world history most if it had gone the other way?

I'm not talking just boxing history.

Which bout, if it had gone differently, would have had the biggest impact on world history?

 

Since we're in alternate history mode --

A Nazi victory in Louis / Schmeling II probably doesn't change much. It was a symptom of international tensions rather than a cause...though it might prolong the color line in US sports by taking some of Louis's shine off.

Burns winning the Johnson bout would be interesting. It would stop the backlash against Johnson that helped keep American sports segregated and (further) poisoned race relations in the United States. If Burns later defended (and lost) against somebody like Langford, you might have seen a small but perceptible improvement in American race politics.

By the time we get to Jeffries/Johnson, it's probably too late to change much. If anything, a Jeffries victory probably cements the color line and further demoralizes the African American community. Not much of a change, and mostly for the worse.

Ryan beats Sullivan -- This one would strangle professional gloved boxing in the cradle. Oh, Queensbury rules would probably win out in the end, but it would be a much harder slog. Boxing would stay a fringe sport, and the first sporting (not boxing, sporting) superstar would never develop. This, in turn, would alter the landscape of American sporting culture during the big leisure boom in the 1880's-1890's. Boxing was pretty culturally significant during this period (especially as a display / symbol of masculinity). Eliminate it, and you see a slightly different Gilded Age culture. As a sidenote, boxing is also known to inflict brain damage, so eliminating Queensbury rules this early would probably stop the loss of one or two IQ points here and there among amateurs. Not a big difference, but a tiny bit.

It's a little cheap, but: Idi Amin suffers serious injuries in one of his fights. Uganda's a lot better off. Africa doesn't have to contend with a public relations nightmare, either.

Ali wins the FOTC. Not a big difference here, but the American civil rights and antiwar movement gets a slightly stronger voice (for good or ill, considering Ali's divisiveness) during the 70's.

By the same token -- Liston knocks Clay out. Malcolm X can't use Clay as a spokesman, which weakens his movement slightly. Maybe enough to give King a greater edge and change the inflection of American civil rights rhetoric during this period.

Molineaux beats Cribb: Nothing happens, probably. Molineaux's not really in a position to do anything about the slave trade, racial perceptions in the UK, etc. It's really too minor, despite its supposed historical significance.