I FORGOT HOW HE ALWAYS CORRECTS OTHER PEOPLE’S MISTAKES.
NO. NO. NO.
Except for when John seemed to be asking that first evening at Angelo’s.
I think Sherlock likes to pretend they’re a couple, and he likes to let everyone else think they’re a couple… but he doesn’t seem to think they could actually be a couple. And it’s not because of any questions of John’s sexual identity or his girlfriends or anything like that. He likes John, right from the start, but he also warns him away, right from the start. He doesn’t see himself as human relationship material. ‘Married to my work’ and all that (but flattered by your interest).
I think ivyblossom said it best in her beautiful fic The Progress of Sherlock Holmes:
Or, as my mother said, “You’ve decided you’re too independent to be loved.”
This sentiment is echoed in the alluded-to but never-shown conversation between Sherlock and Mike Stamford in which he apparently remarked that no one would want to live with him - he’s well aware of how different he is, and he’s by now completely removed himself from the thought of even attempting any sort of meaningful relationship with another human being. He’s resigned himself to it and has even come to revel in it, in antagonizing others and demonstrating just how freakishly different he is.
And then John comes along.
John who thinks he’s brilliant, John who thinks he’s witty, John who wants to know if he has a boyfriend, John who runs through London with him, and John who kills for him. John who doesn’t think he’s a freak, and who doesn’t run away.
Sherlock’s really never known anyone like him, and he wants to keep him, often despite himself. And so while Sherlock likes to let other people think that they’re a couple, that John is his and he is John’s, and while he gets surly and pouty and even catty whenever met with the subject of John’s girlfriends, much less the ladies themselves, he still never quite comes out and says it explicitly, not to anyone they meet, and certainly not to John.
Because this could practically be the Homles family motto:
“All lives end. All hearts are broken. Caring is not an advantage.”
And in Sherlock’s own personal experience, everyone leaves in the end. Everyone gets tired of trying to keep up him and trying to deal with him. Everyone, eventually, sees him for the utterly unloveable freak that he is.
All it takes is a little nudging in the right direction, and John, John who thinks he’s brilliant and witty and worth killing for, calls him a machine, and storms out.
So, no, Sherlock doesn’t correct anyone who says they’re a couple.
But he also doesn’t correct John when he says they’re not.