If you are interested in teaching, NUPOC has an option to fulfill your wishes.
NUPOC offers 4 types of jobs - 2 operational jobs (submarine officer and surface warfare officer) and 2 non operational jobs (power school instructor and nuclear reactors engineer).
As stated before, you major doesn't exclude you from eligibility. If you've taken Calc and calc-based physics you're good (provided you're a US citizen within 30 months of graduation).
As an instructor, after you earn your commission you'll spend 4 years in Charleston, SC teaching, first the enlisted students, then officers. You'll receive some initial training in order to get you ready to teach the way the Navy wants you to teach. You'll teach a variety of subjects including chemistry, math, physcis, reactor physics, thermodynamics and more.
After your 4 year commitment, you either separate from the Navy or try to transfer into another Navy community. I've heard a recruiter say that many in the instructor position choose to go to grad school part-time because it's relatively easy since you'll be in Charleston the whole time (no sea duty) and work hours are pretty regular. But those are words from a recruiter so take make of it what you will.
I've done plenty of research on NUPOC so if you have any questions feel free to PM me. But I will echo what others have said. If your goals are to go into research then I don't see how NUPOC fits in. Unless you want to do research related to Medical Physics, then MAYBE.....and MAYBE getting some teaching experience will help in your goals to become a professor. But realistically the best way to achieve your research and professorship goals are to stay in academia.
While the NUPOC program may sound good, along with the post 9/11 GI bill that comes along with it, you won't NEED it for grad school since most science and engineering PhD programs pay the students' tuition. I mean, I think NUPOC is definitely a good deal (who doesn't want to get paid to go to school?). But first think about what your goals are and then see if NUPOC still fits in.