I believe I read somewhere that while a strong sciences backround helps on the MCAT and USMLE Step I, performance in practice (presumably, though I don't recall, meaning your patients think you're a 'good doctor', among other things) is more based on prior education in the humanities.
But I would venture to say there are loads of great doctors who weren't pre-med as undergrads. Of course, if you mean "a great doctor without going to medical school"... well, that's a horse of a different color, but it apparently worked just fine until the early 1900s.
More on topic - medical professionals who engage in even mid-level (pseudo-)independent practice tend to have master's degrees. PAs, NPs, etc. If paramedics were looking for an increased scope of practice, it stands to reason they would need additional education to handle the vastly increased responsiblity. Even assuming you're willing to accept a purely didactic and clinical program on a technician, rather than technologist, level (without any of that fancy book learnin', especially in 'useless' subjects like the liberal arts), maybe, possibly, an associates degree would be sufficient, especially that were the minimum before beginning clinical work.
Anyway, this has been clobbered to death multiple times in this forum, so I'll shush.