I work a lot on Newton, although it must be confessed that I work in Intellectual History, so I don't really think much about people's lives, instead I think about their ideas. That being said, I've never heard anybody seriously suggest that Newton and Fatio's relationship was sexual. It is often remarked that Newton was a jerk, a weirdo and a loner. While the weirdo part is perhaps too harsh (he was committed to his work), he absolutely was a jerk and (probably because of this) was also a bit of a loner.

This has led some people (including some historians who should know better) to try to explain these things by invoking things like Newton having autism/aspergers, or being gay. He certainly may have had autism or aspergers, but there's no way for us to know. Similarly, he certainly may have been gay. He definitely had no interest in women. But as far as I know there is no actual evidence pointing to Newton being gay, or having a sexual relationship with Fatio.

In Frank Manuel's A Portrait of Issac Newton, he does present that Newton had a very strong attachment to Fatio. The biography, which takes a psychoanalytical examination of Newton's life, Newton's very close relationship with Fatio, and his later breakdown. You see, during Newton's long visits to London, he had built up one of the closest relationships he ever had with the 25 year-old Nicolas Fatio. During this time, they discussed writing a book together, and Newton mentored him on mathematics. After returning to Cambridge, Newton sent many letters requesting that Nicolas come live with him. He promised Fatio money and comfort, but Fatio asked Newton to explain himself plainly (what this exactly means depends on intimation). Whatever was Newton's response, it did not motivate Fatio to relocate, and shortly thereafter their relationship abruptly ended. Then that September, Newton had a mental breakdown, and sent frantic letters to his English friends, some containing indications of Newton's romantic turmoil. Manuel does not go so far as to suggest that Newton and Fatio in fact had carnal relations, but he does suggest that psychologically Newton experienced deep feelings for Fatio that were rebuked when he tried to bring his friend to Cambridge.

If I were to make anything of these kinds of interpretations, I would think that there is probably some kind of truth to this. I do think the kind of explosive relationships Newton "enjoyed" with his peers in England and on the continent were not because he was necessarily proud or narcissistic (though he ended up being a big jerk, as /u/ManicMarine suggests), but because he really thought deeply about his identity and relations with other people. He was extremely hesitant to publish, as many people know, because he vehemently suffered (himself and others) when criticized. I could see him developing some BFF to BF kind of feelings for Fatio, and Fatio being a young, charismatic, opportunistic kind of guy that he was, reciprocating to gain the friendship of a Europe-famous mathematician. While we may never know what happened under those course bed linens in London, I do think it's reasonable to believe that for a time Newton's relationship with Fatio was the most intense relationship of his life (though Hooke might be up there too, lol).

To add to the comments below, it is easy to misinterpret sources of this time. Newton was single, but it was quite normal- even expected- for Oxford or Cambridge dons to be unmarried ( for one thing, marriage implied a decent income to support a wife and a separate household, and children, which could be difficult). Friends would often write very emotional letters to each other, using the word "love" freely. And men would often share beds and living quarters- putting up at an inn would often grant you only space in bed, not a whole bed. Until there's really good straightforward evidence appearing, you just have to shrug.

By the way, I have never heard anyone comment on how truly strange it is, in the Baroque Cycle, that all these late 17th century Europeans seem to have the almost-nil religious lives of 21st century urban Americans. All those courtiers at the later court of Louis XIV were, in between dalliances, reading books of sermons and arguing over their fine points.