The result itself seems to be due to A. Hurwitz: 
"Uber algebraische Gebilde mit eindeutigen Transformationen in sich," Math. Ann., 41:403–442, 1893. 

At least, this is what Babai refer to on page 42 [here](http://people.cs.uchicago.edu/~laci/handbook/handbookchapter27.pdf), as well as 
Macbeath on page 106 [here](http://library.msri.org/books/Book35/files/macbeath.pdf). 

If you read German, you should take a look at the paper and check if this is a correct reference and if Hurwitz used arguments similar to the ones appearing in the proof of Lefschetz fixed-point theorem.  

Sorry, in my earlier answer I misunderstood your question.