Even the attitude irks me. You need to fire someone, you fire them, you don't "make a public example" of them as a sign of obeisance to the head of a competing company. I want to know my boss has got my back, rather than thinking they're willing to throw me under the bus without so much as asking me about it if they get a harshly worded email from someone else.
Just for your information, I've been on three separate management courses which say the contrary - when you're trying to set a cultural norm in your company, making a public example of infringers is practically considered the textbook response. The example given usually concerns sexual harassment. You make the punishment of the harasser public as a way of reinforcing that you're serious about the issue.
That said, most management courses will also tell you to privilege process over people - if something goes wrong it's because the process failed, not the person, and as such most of the time a public correction of someone making a mistake is not appropriate, unless it is felt that the person acted knowingly and deliberately against the rules... So unless the recruiter in question had already been corrected on this error, not only should they not have been fired, but they should not even have been reprimanded in public.
I've been on three separate management courses which say the contrary - when you're trying to set a cultural norm in your company, making a public example of infringers is practically considered the textbook response.
Then the textbook is wrong and should be thrown out. These are people we're talking about, real, actual, flesh-and-blood people... with feelings, family, friends, lives, hopes, dreams, etc. Not fucking "resources" or some fungible asset that can be treated as nothing more than a cell in a spreadsheet, and certainly not something that is a valid target for public shaming.