In the US, "medical school" usually refers to much higher level education - the general assumption is full-fledged doctor or surgeon. While those aren't the only professions that go to med school by any means, it's a much longer and more expensive process than most any other educational path bar law. It's definitely not a high school thing - pre-med alone is generally 2 years of university-level work, and that's before you can even apply for med school proper in most cases.
RNs (registered nurses, as opposed to more specialized things like nurse practitioners or certain kinds of psych nurses) generally go to more traditional universities in the US. It's still hardly easy, but it's a whole order of magnitude off of what we call "med school", and it's a different program entirely rather than a "leveling up" system from nursing up to doctoring.
"Medical high school" is to my knowledge (which, granted, is limited, since I'm not in medicine myself) unheard of, as education for minors is usually very broad and general. There are special preparatory schools - private schools that try to prepare students for university-level work, which often have college-level courses and a handful of college credits built in - but none of them are going to count as a substitute for a university education at all.