I am currently an undergraduate math major and have hopes of attending graduate school in applied math. I also have a love for physics and thinking of minoring in physics. After some research, I found that many departments have professors in the research area of mathematical physics. Is having a minor in physics a good background to pursue research at the graduate level in mathematical physics?
It depends on what courses you end up taking as part of your physics minor. Mathematical physics leans heavily on Lagrangian and Hamiltonian mechanics, quantum mechanics, statistical mechanics, and electrodynamics.
I don't know how many credits you have available for your minor and what the requirements are in your university to take those courses. At the very least, make sure that you have taken both classical mechanics and quantum mechanics. Your math background in real and complex analysis will put you in great shape to learn the formalisms of the other theories.
Degree names don't matter: coursework matters
It really depends on what subjects you've taken courses in and what subjects you've learned in your free time. Grad schools don't particularly care if your degree label lines up with what they're offering, they care if your coursework and professional experience does. In the US, many graduate programs explicitly list the courses they want you to have as a minimum background. Contact their admissions office or look through their websites for the required background coursework.