Ramos gave her a third violation, which resulted in a game penalty, putting Osaka 5-3 ahead. A tearful Williams argued her case with tournament officials but, although she held serve in the next game, Osaka served out the victory 6-2 6-4.
Here is what happened, a full transcript of exchanges on court and how tennis reacted.
How the incident unfolded
Williams, having lost the first set to Osaka 2-6, was handed a coaching violation early in the second after her coach Patrick Mouratoglou made a hand gesture towards her from the crowd.
She smashed her racket after dropping serve at 3-3, leading to a docked point for a second violation.
Umpire Ramos gave her a third violation after Williams accused him of being a 'thief', which resulted in a game penalty, putting Osaka 5-3 ahead.
A tearful Williams argued her case with tournament officials but, although she held serve in the next game, Osaka served out the victory 6-2 6-4. Williams refused to shake hands with Ramos after the match.
The trophy ceremony was met with a chorus of boos from Williams' home US Open crowd, who had been unhappy at her treatment during the match. Williams urged the crowd to show their support for Osaka following her first grand slam victory.
Full transcript of what was said on court
Second set, Serena Williams 1-0 up:
Mouratoglou appears to make gesture during the first game, deemed by the umpire to be in-game coaching.
Williams to umpire Ramos:
"If he gives me a thumbs up, he's telling me to 'come on'. We don't have any code, and I know you don't know that. And I understand why you may have thought that was coaching, but I am telling you it is not. I don't cheat to win, I would rather lose. I'm just letting you know."
Second set, Serena Williams 2-1 up:
Between games, Williams and umpire Ramos have another exchange:
Williams: "I can see where you may have thought that... I have never got a coaching violation, because I don't do it. It's not something I do. "I can understand why you may have thought that but just know I've never cheated.
Ramos: "I understand your reaction as well"
Williams: "OK, thank you. Because I don't cheat.
Ramos: "I know that"
Williams: Yeah, so thank you so much.
Second set, Serena Williams 3-2 up:
Williams smashes her racket after losing the fifth game, handing her a second violation and a point penalty.
Before the sixth game begins:
Williams: "This is unbelievable, every time I play here I have problems. That's a warning. I didn't get coaching, I didn't get coaching. I didn't get coaching. You need to make an announcement I didn't get coaching.
"I don't cheat, I didn't get coaching. How can you say that? You owe me an apology, you owe me an apology. I have never cheated in my life. I have a daughter and I stand for what's right for her and I have never cheated. And you owe me an apology."
Second set, Naomi Osaka 4-3 up:
In the break between games, Williams and Ramos have another exchange:
Williams: "For you to attack my character, something is wrong. It's wrong. You attacking my character, yes you are. You owe me an apology. You will never ever ever be on another court of mine as long as you live. You are a liar.
"When are you going to give me my apology? You owe me an apology. Say it. Say you're sorry. [umpire inaudible] Then don't talk to me, don't talk to me.
"How dare you insinuate I was cheating.
"You stole a point from me, you're a thief too."
Ramos to crowd: Code violation. Verbal abuse. Game penalty, Mrs Williams.
Confusion follows as Osaka lines up to serve, when post-penalty it should be Williams. This pause leads to another exchange between Ramos and Williams.
Williams: Are you kidding me? Are you kidding me? As I've said, you're a thief because you stole a point from me. But I'm not a cheater. I told you to apologise to me. This is out... excuse me I need the referee.
Tournament referee Brian Earley arrives on court.
Williams to Earley: "This is not fair, this has happened to me too many times. This is not fair.
"To lose a game for saying that, it's not fair. How many other men do things? There's a lot of men out here who have said a lot of things. It's because I am a woman, and that's not right."
"Because I'm a woman, you're going to take this away from me? It's not right."
"I know you can't admit it, but you know it's not right. You can't change it, but I'm just saying...
"I get the rules, but I'm it's not right. It happens to me at this tournament every single year, it's not fair. That's all I'm saying."
The game penalty stands. Williams is 5-3 down in the second set, but holds her serve without dropping a point to make it 5-4.
Again Williams talks to the tournament supervisor.
Osaka holds serve to win her first grand slam to a mix of applause and jeers for the umpire. Williams refuses to shake the umpire's hand.
What the rules say
From the Official Grand Slam rulebook 2018:
Players shall not receive coaching during a match (including the warm-up). Communications of any kind, audible or visible, between a player and a coach may be construed as coaching.
"I was coaching but I don't think she looked at me," Williams' coach Mouratoglou said afterwards. "I am honest, I was coaching, Sascha [Bajin, Osaka's coach] was coaching the whole time too. This is one of the rules that is ruining tennis. She [Williams] will struggle to come back from this."
'I'm here fighting for women's rights' - what Williams said afterwards
"I can't sit here and say I wouldn't say he's a thief, because I thought he took a game from me.
"But I've seen other men call other umpires several things. I'm here fighting for women's rights and for women's equality and for all kinds of stuff. For me to say 'thief' and for him to take a game, it made me feel like it was a sexist remark. He's never taken a game from a man because they said 'thief'. It blows my mind.
"I just feel like the fact that I have to go through this is just an example for the next person that has emotions, and that wants to express themselves, and wants to be a strong woman.
"They're going to be allowed to do that because of today. Maybe it didn't work out for me, but it's going to work out for the next person."
Responding to her coach's admittance he had been coaching her, Williams said: "I literally just heard that too. I just texted Patrick because we don't have signals, we've never discussed signals. I want to clarify myself what he's talking about. I wasn't being coached."
How the tennis world reacted
There were two statements from the US Open:
The WTA confirmed they would "look into" what happened.
A statement read: "There are matters that need to be looked into that took place during the match. For tonight, it is time to celebrate these two amazing players, both of whom have great integrity.
"Naomi is a deserving champion and Serena at all times plays with class and makes us proud."
'Williams is calling out a double standard' - Billie Jean King
"Several things went very wrong during" the match, King wrote. "Coaching on every point should be allowed in tennis. It isn't, and as a result, a player was penalized for the actions of her coach. This should not happen."
In a second tweet, King said: "When a woman is emotional, she's 'hysterical' and she's penalised for it. When a man does the same, he's 'outspoken' & and there are no repercussions. Thank you (Serena Williams) for calling out this double standard. More voices are needed to do the same."
'Umpire was too quick to penalise' - Chris Evert
"Because of the big occasion - finals, the score, a game penalty - he should have warned her," the American told BBC Radio 5 live.
"Scold her. 'Ms Williams you need to be quiet because if you keep going on like this it will be a game.'
"Because of the enormity of the moment he should have given her a little bit of a break - but instead he just went right for the jugular."
'She wasn't even looking. Believe what you want' - US Open quarter-finalist Mardy Fish
"What a wild US Open for the Ref's [sic]. Two ridiculous calls today. I can promise you, that's not coaching, racquet abuse no doubt, but the verbal abuse??? It's the US Open Final!!!"
'You just can't act like that' - Andrew Castle
"Not sure how any unbiased observer who knows the rules and history of tennis can look at what happened and defend Serena," he tweeted.
"Memories of Clijsters match. You just can't act like that I'm afraid. #Serena now claiming that men do this. More nonsense. Is she claiming sexism? This is not right."
'You can't talk to people like that' - Liam Broady
"I think incredibly strong from the umpire to not be intimidated by a GOAT (greatest of all time) of the game and hand out the game penalty even so?
"You shouldn't talk to anybody in this way whether they're an umpire or person on the street."
'Ramos must be defended' - Richard Ings, a former rules and competition chief for the ATP
"When coaches and players come out and threaten the employment of umpires then it falls to the governing bodies to defend those officials just doing their job," he tweeted.
"Carlos Ramos was doing his job. And doing it well in my extremely experienced opinion."
What is next?
The Women's Tennis Association says it is going to look into 'matters' from the US Open final, suggesting both Williams' reaction and whether Carlos Ramos was too quick to issue a violation.
But the wider tennis debate will centre on whether men and women are treated differently. One way of ending the perceived discrepancy around in-game coaching would be to make it legal, therefore removing the grey area of its definition.
