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Williams, who was seeking a record-equalling 24th Grand Slam singles title on Saturday, was handed a warning for a coaching violation before being deducted a point for smashing her racquet.
She then had a heated argument with chair umpire Carlos Ramos, which cost her a game.
The six-times US Open champion, who has since been fined $17,000 by the United States Tennis Association for the violations, vigorously disputed each during the match.
In the wake of Osaka's first Grand Slam triumph, there were messages of support for Williams as well as those condemning her behaviour and agreeing with the umpire's calls.
Tennis great Billy Jean King wrote on Twitter: "When a woman is emotional, she's "hysterical" and she's penalized for it. When a man does the same, he's "outspoken" and there are no repercussions. Thank you, @serenawilliams, for calling out this double standard. More voices are needed to do the same."
Yet Australian Margaret Court, whose tally of Grand Slam singles titles is being chased by Williams, had little sympathy for the 36-year-old American former world number one.
"We always had to go by the rules," Court, who dominated tennis during the 1960s and early 1970s, said according to a report in The Australian.
"It's sad for the sport when a player tries to become bigger than the rules.
"Because the young player outplayed her in the first set, I think pressure got her more than anything."
The drama started when Ramos handed Williams a coaching violation early in the second set because of hand gestures made from the stands by her coach Patrick Mouratoglou. He later admitted the offence, which that is not allowed in the sport but rarely enforced.
When the violation was announced Williams approached Ramos to insist she never takes coaching and would rather lose than "cheat to win".
Things seemed to settle down as Williams went on to break Osaka for a 3-1 lead, but she gave the break right back in the next game with a pair of double faults, prompting the former champion to smash her racquet on the court.
That resulted in a second violation, meaning Osaka was awarded the first point of the sixth game.
Williams, who was under the impression the first violation had been rescinded, returned to Ramos to seek an apology for saying she had received coaching earlier.
During a changeover, Williams resumed her argument with the umpire, this time saying he was attacking her character and was a "thief". That triggered a third violation, which resulted in a game penalty that gave Osaka a 5-3 lead.
From there, Williams summoned the tournament referee to the court and said male tennis players are not punished for similar offences.
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.
According to McEnroe, Ramos should not have given Williams a violation for breaking her racquet and should have warned her early on about what would happen if she did not move on.
"I've said far worse," McEnroe, a seven-times Grand Slam singles winner, said on ESPN. "She's right about the guys being held to a different standard, there's no question."
Yet Richard Ings, a former professional chair umpire who also used to be the ATP Tour Executive Vice-President, Rules and Competition, felt it was Williams who needed to apologise.
Ings once issued a warning, point penalty and a game penalty against McEnroe at the 1987 US Open for obscenities directed at the umpire.
"We should not let her record, as glowing as it is, overshadow the fact that on this day, in this match Williams was wrong," Ings wrote in The Sydney Morning Herald.
"The decisions made by Ramos had nothing to do with sexism or racism. They had everything to do with observing clear breaches of the grand slam code of conduct and then having the courage to call them without fear or favour."