What do we know about the fight between Salazar Slytherin and Godric Gryffindor?
We know that according to professor Binns, it was because Slytherin felt Muggle-Born students to be "untrustworthy".
Has JK Rowling commented further on the subject?
Furthermore, has she mentioned whether this will be addressed in Fantastic Beasts?
The exact details of everything that happened leading up to Salazar Slytherin leaving Hogwarts aren’t all known, but in Harry’s fifth year, the Sorting Hat tells a bit more about it.
“And never did they dream that they
Might some day be divided,
For were there such friends anywhere
As Slytherin and Gryffindor?
Unless it was the second pair
Of Hufflepuff and Ravenclaw?
So how could it have gone so wrong?
How could such friendships fail?
Why, I was there and so can tell
The whole sad, sorry tale.”
- Harry Potter and the Order of the Phoenix, Chapter 11 (The Sorting Hat’s New Song)
Though he doesn’t give that many details on what the fighting was about, he does give some more information about it. He says that when the relationships between the founders became more tumultuous, each of the Hogwarts houses began vying for control of Hogwarts. He also refers to there being dueling, which means there were likely several duels among the founders in that time. Then, he says Salazar left one morning, and the fighting died out after that.
“So Hogwarts worked in harmony
For several happy years,
But then discord crept among us
Feeding on our faults and fears.
The houses that, like pillars four,
Had once held up our school,
Now turned upon each other and,
Divided, sought to rule.
And for a while it seemed the school
Must meet an early end,
What with duelling and with fighting
And the clash of friend on friend
And at last there came a morning
When old Slytherin departed
And though the fighting then died out
He left us quite downhearted.”
- Harry Potter and the Order of the Phoenix, Chapter 11 (The Sorting Hat’s New Song)
The Sorting Hat never specifies what the fighting was about, and doesn’t say exactly what made Salazar choose to leave. Though he mentions that the founders all had different preferences in the students they admitted, he never said that the core of the fighting was that Salazar didn’t want to admit wizards born to Muggles while the others did. This may still be the reason, but it’s also possible that the true reason is something else, and Professor Binns is incorrect. The Sorting Hat would know what the true reason was, having been there himself, but since he doesn’t state it, we don’t know what it is.
In addition to the excellent information in Bellatrix's answer, I want to add a few more details that we know: (all non-book quotes are from the Chamber of Secrets article on Pottermore)
Salazar Slytherin had a grandiose opinion about his importance to the school:
However, it is clear by the very decoration of the Chamber that by the time Slytherin finished it he had developed grandiose ideas of his own importance to the school. No other founder left behind them a gigantic statue of themselves or draped the school in emblems of their own personal powers (the snakes carved around the Chamber of Secrets being a reference to Slytherin’s powers as a Parselmouth)
In addition to the question of taking in Muggle-Borns, he also disagreed about whether to teach the Dark Arts.
Perhaps, when he first constructed the Chamber, Slytherin wanted no more than a place in which to instruct his students in spells of which the other three founders may have disapproved (disagreements sprung up early around the teaching of the Dark Arts).
While Professer Binns and the Sorting Hat merely say he left or departed, seemingly of his own accord -
After a while, there was a serious argument on the subject between Slytherin and Gryffindor, and Slytherin left the school.” (COS)
And at last there came a morning When old Slytherin departed (OOTP)
Later canon seems to say he was forced out.
What is certain is that by the time Slytherin was forced out of the school by the other three founders,
Also, while Professor Binns says that the fight was between Slytherin and Gryffindor, and the Chamber of Secrets article says the fight was between Slytherin and the remaining three founders, the Sorting Hat says they all fought with each other (even Hufflepuff and Ravenclaw fought with each other):
“And never did they dream that they Might some day be divided, For were there such friends anywhere As Slytherin and Gryffindor? Unless it was the second pair Of Hufflepuff and Ravenclaw? So how could it have gone so wrong? How could such friendships fail? Why, I was there and so can tell The whole sad, sorry tale."
Considering that the Sorting Hat was there at the time, it seems likely that his version is correct, and the fight was between all the founders, and that Slytherin may have actually left of his own accord.
Furthermore, the Sorting Hat (deliberately, based on the aftermath of the song) seemed to imply that all the founders were equally at fault for the fights, unlike Professor Binn's version, and the one at Pottermore, which imply that Slytherin bears the main brunt of the blame.
