I wish we could have a new version of this question made, not just because this question is old, but because I think you can absolutely have a unicorn.
But it's not a horse
These things show up in several medieval bestiaries. At one point it was included in the Scottish coat of arms. but look at these depictions. Often depicted with a billy goat style beard and cloven (split) hooves.
Now why was it included in the Scottish coat of arms? Because it was said to be one of the only animals to be able to kill a lion. And lions happen to be the national animal of England. Nice little rivalry there. But that is odd, lions live so far away. They live in Africa!
Now, what lives in Africa where lions live, and potentially has horns and cloven hooves, and might even sport a billy goat style beard?
Now, is it possible that there was ever a species that had horns that grew so closely together that they fused into one horn? Or if not a whole species, a disorder that might cause it to happen in certain individual antelopes? I think, ABSOLUTELY!
Could this animal through repeated depictions be later confused for a horned horse? Again, quite possible.
And I would be remiss if I didn't point out that there are said to be a few medieval accounts of people actually hunting for them, meaning they may have been real animals hunted to extinction at some point in the past. But I can't seem to find the article that actually listed those sources, so take that with a grain of salt. In absence of any actual historical sources to show you, I might just say that the possibility of it being real is there, and biologically speaking, there's no reason they couldn't have existed.
So, could they exist? Yes. But did they really ever exist? Well, that's a mystery we may never know for sure.