This is so shortshighted, it's not about 'respect', its about market value.
By bringing an offer your employees are proving their market value, and you don't want to match it, so they will leave.
Leaving is exactly what JohnFen actually recommended to these employees.
I just choose not to play that game. It's unnecessary. If an employee can't just come to me and be straight about their compensation requirements, that's a problem.
If someone asked me for a pay raise and I turned it down, I'd also explain exact why I turned it down. (This is hypothetical as I've not turned down a pay raise). I wouldn't think poorly of anyone for just asking for a raise, but I suppose I'd start to get annoyed if they did it daily when the reason that I turned them down was still true.
That's exactly what they are doing, but with evidence to boot so you are on the same page.
Having an offer in hand is valuable to the employee, certainly! There's nothing wrong with getting an offer, then asking your current employer for a raise that would meet or exceed that offer. The issue I have is actually communicating that offer to the current employer.
Now, don't get me wrong -- I'd certainly never punish or fire anyone who did that. I'm just not going to give in to what I perceive as extortion, and what I perceive as being unfair to the employer who made the offer.