Spidey was never written that way.
Yet. So why not start with a movie? If the idea takes off, then you can bet they'll do a Spider-Girl-esque alternate continuity title where Spider-Man is black. Besides, there is that famous comic book scene where the little black kid imagines that Spider-Man simply
must be black...
The comic book industry has huge issues with creating and promoting basically any non-white-male character. Filtering the characters in from other media may be the only way to kick-start the process of de-whitewashing the two core superhero universes.
Also, we just had a series of flicks depicting a fairly accurate white Peter Parker, so changing his race might be a good way to distinguish the new line of flicks from the old one. No point in basically making the same movie twice, right?
(Yeah, I know Hollywood is addicted to remakes, but the remakes that are any good change tend to change something around from the original flick so the remake is at least slightly its own story.)
However there's no need a turn a major character into a different race.
There
is if the actor who best fits your vision of the character-- or the actor who is the most
bankable candidate for the character-- happens to not be white. If they don't change anything else about the story besides making the Parkers a black family, who cares?
I could see changing Peter Parker into an Asian or Hispanic character perhaps being more difficult, but there's nothing about Peter's basic backstory that disagrees with the sort of lives young black kids tend to lead. There's nothing about the setting that makes it hard to work, either.
Why not make a Black Panther movie? Why not put Vixen in a film? Where is the Luke Cage film that they were working so hard on?
A Black Panther movie would be cool. Vixen is kind of a boring and stupid character, they'd have to cast her with one hell of an actress to get me to see it. Luke Cage is so mired in various stereotypes about black culture that I think he'd only work in the context of a Power Man & Iron Fist movie that would force them out of a lot of those stereotypes.
It also wouldn't make sense to change the race of certain characters. For instance Black Panther can't be white (because he is the leader of a African nation), and Captain America can't be black (because in 1942 American wouldn't give a black man the mantle of their Propaganda cheerleader).
You
could do a black Captain America, you'd just have to change around a lot of the basic story details to accommodate the character's blackness. Yeah, he couldn't be a propaganda cheerleader that way, but you could still do another depiction of the character.
Draw on the mini-series that Patriot's backstory comes from, for example, and turn Captain America's story into an assault on old-fashioned racist beliefs. Depict Cap as more of an espionage sort of character, a bit more hard-edged, in line with the Brubaker version of the character.
I think there could be a certain satisfaction to a successful depiction of Cap as a black man. Remember how everyone felt when Jesse Owens took the gold in Berlin in the 30s? Now imagine a black man single-handedly routing Nazi battalions and punching Hitler in the face. That could be daring, interesting stuff.
A white Black Panther would be The Phantom more or less exactly. You could do the story, but you'd have to handle it really carefully to keep the narrative from seeming ridiculously racist.
That said, Black Panther is probably more marketable as a black character just because the character is honestly more interesting that way. He's about the only "jungle hero" who can ditch the horrible "white man's burden" racial baggage associated with the archetype, simply by virtue of being the only black take on it.