From my end of things, the only acceptable reason for him to be called "the black doctor", if he's the only black actor that ever gets the role. And even then, I'm gonna be dubious of whoever said it until I discover that they're a new fan/viewer.
See, it's a long standing tradition to refer to a given era of the show by the last name of the actor. Gatwa isn't difficult to pronounce, so only a noob to the show would think it was necessary to use the race of the actor to narrow things down. Well, noobs and racists.
It's the same with Whittaker. There's never been a Whittaker as the doctor before, so why would anyone bother with that as the first way to denote which era it was?
Now, in twenty years, Whittaker might be called the first female Dr Who. And Gatwa might be the first black Dr Who, but until then, then being the only actors in the role with their names is plenty for anyone that's a fan.
Also, I'm still wishing there had been an Elba Bond era. I was so hyped when the rumors he was maybe going to do it. I really think he would have reinvented the role in a way few actors could. The way he moves, and his subtle acting style combined with his voice would have been at least as powerful a screen presence as Connery. Different, for sure, but I genuinely believe the role would have become shaped by him in a way that nobody since Connery and Moore pulled off.
Anyway, I've never seen anything Gatwa has been in, but I'm always a little hyped for a new doctor, no matter how much I enjoy the existing one. As long as the writers don't screw things up, and the showrunner doesn't do horrible things, Gatwa is at least as interesting as Smith was coming in as the youngest doctor, for me. I had never even heard of Smith before he got the role (unlike all the others in the modern doctor). Here's to Gatwa being the doctor for years and years!