What Scantee and Acroy said.
1. Totally unprofessional. Can you figure out a way to file a complaint with management? The company could have liability if its employee is broadcasting private records around. Or maybe that's your response the next time she starts into it -- "Wow, X, I never realized Company Y's HR policies allowed people to talk about employee records outside the office -- I'd totally get fired for that. Can you forward me the policy? My management is a huge PITA about employee privacy, and I'd love to be able to convince them that they're being overprotective."
2. Yes, long-term partners shouldn't tear down each other behind the partner's back. But for many people, it is also sort of presumed to be ok to vent or joke about certain foibles. I have certainly been known to kvetch that my brilliant engineer DH, who is so insistent on efficiency in all aspects of his life, still cannot figure out that the round tupperware go with the round tupperware and the square ones go with the square ones. And sometimes you might blame your partner to save face or fit in with the group -- e.g., maybe he gets asked to go out to lunch, but he is trying to cut back, so he says, "aw, man, love to, but X wants me to cut back." It's immature, yes, but it's not malicious. Or, heck, maybe he even mentioned it in a totally off-hand way -- "man, I've been avoiding lunches out for two weeks now, and my bank account thanks me" -- which led into a discussion of how you helped him through it, which in turn was twisted by someone with less-than-charitable intentions. Or maybe she thought she was making a joke, and it fell flat. Intent matters -- in fact, I'd say it's the only thing that matters.
Side question is whether you are upset that he said something, or upset that he said something to this particular person, who sounds like a real treasure. Again, two different issues, so make sure you're addressing the right one.
So, yes, good to talk to him about it -- but I'd recommend approaching it from the angle of wanting to understand the circumstances first, before jumping to the conclusion that he was intentionally disrespecting you/airing private laundry to someone you despise. And if this turns out to be an area where you guys have different assumptions of what is appropriate with third parties, talk that through, too -- don't just assume that your approch is right and his is wrong and so he must therefore conform to your expectations. Figure out why this particular issue got to you, and then figure out how to draw lines around what is ok and what isn't that acknowledge his periodic natural need to vent.
And FWIW, the best way to deal with malicious people is never, ever to respond in kind -- that just lets them know they have the power to get to you. Think of her as a gnat flitting around your head: annoying, but not even meaningful enough to take a swat at. If you're going to spend time thinking of rejoiners for the next time, think of things like taking her comments completely literally ("oh, yes, charity is very important to both of us, I have been SO impressed by [partner's] commitment to good causes") or extolling the virtues of the thing she is trying to criticize ("oh, yes, I have been SO impressed by [partner's] efforts to cut back -- it feels so great to know we're on-track to retire in only six more years!).
Tl;dr: She who FIREs first always has the last laugh.