"He doesn't like to try new things because it stresses him out."

Has he seen a mental health professional or looked into ARFID, maybe? I only ask because though I'm not diagnosed, I've been exhibiting similar behaviours since childhood and can empathise with the thought of new foods being stressful: I have a limited diet of foods that are "safe" to eat and the thought of trying new things, or things with textures/smells/appearances that my brain is for whatever reason repulsed by, also stresses me out and makes me panicky. I will, however, try new things if they're presented to me in a low-stress environment (i.e, "I'm making/ordering [x food I don't eat], do you want to try a piece?" or "We're going to a Chinese buffet, want to come?") but I absolutely can't if it's presented like a demand ("You don't eat anything! Sit down and eat this thing!")

It's not your job to nag him; you're his partner, not his mother. Only you can decide if this is a dealbreaker for you. I'd suggest, first and foremost, recommend that he see someone about his issues with food and, with them, come up with a plan to tackle or mitigate it. The onus of making him try things shouldn't be on you.