She was disqualified from a semifinal match at the 2009 US Open for threatening a lineswoman.
However, the world's third-ranked men's player does not necessarily agree with the assessments of Williams and WTA chief executive Steve Simon that umpires treat women players differently from men.
But Murray said of the sexism claims: "I think that's a bit far-fetched".
The 23-time Grand Slam champion had another running battle last Saturday with umpire Carlos Ramos, calling him a "liar" and a "thief" after he issued her a code violation for receiving coaching from Patrick Mouratoglou, which she vehemently denied. "Don't worry about me". In comparison, I never saw Nadal shouting like that with an umpire. Williams, clearly unhappy with the ruling, went on to berate Ramos for his judgment, repeatedly demanding that he apologize for branding her a cheater.
"The way Serena handled the situation, I think she handled it with such grace and class".More news: Pope Francis to Meet with US, World Catholic Leaders
In sports, stuff happens and how an athlete copes with that stuff can determine whether he or she ends up with a championship.
That anger would later cost her the game, and ultimately the match - although Osaka was clearly the better player on the night and would've probably taken out the match regardless. "I'm here fighting for women's rights and for women's equality". And for me to say "thief" and for him to take a game?
One suggestion being floated is to refuse any match assignments involving Williams until she apologizes for vilifying Ramos. "But in my mind, I really wanted to know what was going on".
Tennis great John McEnroe, one of the game's most tempestuous characters in his playing days, said the sport must find a way to allow players to express feelings and inject their personality into the game while adhering to certain rules.
Most of those criticisms - largely made by men - were valid, but they chose to ignore Williams' experience as a black woman.
The International Tennis Federation has defended Ramos for his actions during the final.
"It's odd because that's an individual sport, but I got used to it". "But I'm going to continue to fight for women and to fight for us to have equal [treatment]".