Anybody, NP or anyone else, can go to med school if they take the classes required beforehand.
Most NP programs won't have you doing the classes you need to do well on the MCAT, and they won't have been recently. I didn't have to learn physics or organic chem for nursing school, and my local NP programs don't require them either. Some med schools themselves don't technically require you to take all the hard science classes any more either, but if you don't have a competitive MCAT score, you have pretty much no chance of getting in anywhere.
Even the people who took organic, physics, etc. in nursing school probably won't remember it well enough to do well on the chem/phys section if it's been long enough since they've taken those classes. The chem/phys section was a nightmare and I took those things very recently.
Most med schools actually have a two step application process... you apply initially, they screen you for a decent MCAT score, and then you're allowed to complete the secondary application if your MCAT score is good enough. Unfortunately for people with good, competitive healthcare experience, like RNs, NPs, PAs, etc., they don't even see your healthcare experience if you don't pass the primary application stage because your MCAT score sucked. So, experience means absolutely nothing if you don't do well on the MCAT in the first place, which is a broken system if you ask me because they'll lose GOOD, competent people with relevant healthcare experience, but that's not what they're interested in up front.