A comma is not needed in this sentence. Whether or not it's wrong to include it is open for debate (though I would say it isn't). It doesn't feel unnatural, but it's generally good practice to avoid using commas anywhere they aren't necessary. Otherwise you might end up with "comma abuse" and, instead of making your writing easier, the copious amount of commas can actually start to muddle the meaning of your sentences.
FumbleFingers gives you a good rule of thumb in his comment -- "If you wouldn't automatically pause in speech at that point, you shouldn't want a comma in the written form." This won't be correct in every single instance, but is right far more often than it's wrong.
This website has a good list of instances when a comma is required, complete with examples: link
So why does a comma still feel appropriate here?
If you read the page I linked above you'll see the closest category this particular comma might fall under would be "setting off a parenthetical element." Something is a parenthetical element if it can be completely removed from the sentence without the sentence suffering. And it's true that doing so in this case would still leave a valid sentence:
However, I couldn't disprove the existence of flying unicorns or wish-granting fairies.
But the meaning isn't quite the same. Since the sentence begins with "however," we know you're contrasting something. Without "either" it sounds like the previous sentence ought to be something like this:
After a long discussion, I managed to disprove almost everything she'd said. However, I couldn't disprove the existence of flying unicorns or wish-granting fairies.
So I would argue that "either" is very much needed in order to convey the meaning you're actually trying to get across. Chopping it out changes the meaning of the text left behind, even if it doesn't make it grammatically incorrect, and that means it can't be a parenthetical element. It's the fact that it "feels like" a parenthetical element that makes the comma seem natural -- and also what makes me say that it's not incorrect, just unnecessary.
