I asked about the "armpit" (singular) instead of "armpits" (plural) because a friend of mine once had an infected pore in her underarm, and when the doctor lanced it, out came a whole ping-pong-ball-sized amount of liquid, which she said was really awful smelling, like sweat but a lot worse. So it is possible to get a blocked pore that then creates a cyst, and it could indeed stink. Know if what he is dealing with is really only on one side?
The way I see it it is either you tell him or someone else tells him and publicly embarrasses him. I can't tell how long are you dating but it is probably long enough for you to give honest opinions and inputs to each other. I am 7 years together now with my husband, and we can tell everything directly to each other, but we were able to do it pretty early in relationship. I tell him openly when he wears something I dont like, and he sometimes takes the criticsm, sometimes doesn't. If I do something that makes me sweat he usually tells me "Hey you smell sweaty" or he openly tells me if I have a bad breath. And I appreciate that cause I can do something about it before I go out of the house and embarrass myself. I don't think he will be mad at you for telling him, he might get embarrassed, but that is kinda imminent.
Deodorants, on their own, don't prevent sweating, they just reduce odor by limiting odor-causing bacteria. Consider using a stronger over-the-counter antiperspirant such as Certain Dri, Secret Clinical Strength, Gillette Clinical. Apply antiperspirant once in the morning and again prior to bedtime.
I notice you using the singular, "a smelly armpit." Usually when someone has underarm odor it is in both armpits. If you really literally mean one armpit and not the other, it might be a blocked pore or other cause that is more medical than simply hygienic. If that is the case, you could ask him about it without embarrassment -- "Have you noticed that it is sore or swollen under your arm? Do you think you should see the doctor in case there is an infected pore or cyst?" But if you are merely saying 'a smelly armpit' as a general term for both arms, this of course does not apply.
I agree with specialmom. This is a "rip the bandaid off" moment. It's going to be uncomfortable for you to say something, but probably better just to face it head-on ASAP. Buy deodorant and tell him you'd appreciate if he'd use it. He'll be a bit embarrassed and you'll be a bit embarrassed, but only for a few minutes. It's worth investing a few minutes of embarrassment for a long-term better outcome, right?
It's probably hampering his relationship with others, and he doesn't know it, so you're doing him a favor.
So, truth can hurt. But when it is for the greater good, you have to do it. Onion pits are not sexy. And they make others that your boyfriend is around too uncomfortable. What you might do is buy some antibacterial soap and say "I read that washing with this under your arms in the shower is a good thing to do to get it clean and odor free. And then buy him a good deodorant. Tell him you'd appreciate him using it.
You can't have a sincere relationship with someone you are afraid of saying 'dude, the pits. it's bad". I have two sons . . . OMG< the stench at times. And I laugh as I tell them to take care of it. I mean, really. I can't handle it.
Long term serious relationships will have a million moments in which you have to talk about things just like this. Wouldn't you want to know if it were you? Help him out! good luck