Whilst having no medical qualifications I do concur with previous comments. Whilst it was a fair old time between being sick and eating beetroot it's seems possible it's coloured everything down there. And no doubt I'll be told I'm talking nonsense but if it was blood and bright red I feel confident you'd have posted your second post, the next day, from hospital, since it would have suggested new blood to me. I'd have thought old blood would have darkened considerably, maybe even to black ?
But in your shoes I think I'd have annoyed the local A&E at the time, even if just to put my mind at rest, but I think you're past that point now.