While "fire" and "terminate" (someone) might bear too much negative emotions, "asking to leave" leans too much on the rosy side. Usually "asking" implies they have a choice, right?
Fred's further advice of getting right to the point and giving them honest, clear reasons is spot on. He's not advocating for bullshitting the employee; he's advocating for not being a dick when there's no reason for it.
I disagree. Using too rosy a term for something that's very bad for the recipient might be perceived as disingenuous or just lacking balls to tell the unpleasant truth.
Maybe more of a Hobson's choice.
"I'm going to have to ask you to leave" seems to be the sort of thing said by bartenders/bouncers when you've been cut off and you're demanding that one last beer. Or by a police officer who presents you with the choice of walking out on your own, or or getting arrested.
"John, due to reasons X and Y you will no longer be employed with this company."
If I had to choose, I'd rather hear something like that.
"I can quit, or you can fire me. But I'm not quitting."
With the latter, you can qualify for unemployment benefits. With the former, you get nothing. See the difference?
Also keep in mind that there's a difference between what you might say to the employee, and what you subsequently report to others. Fred talks about how you need to be sensitive to how the news affects others on the team. And so you might have a direct conversation with an employee who warrants firing, but you may prefer to tell other employees, investors, or the public that this person was "asked to leave."
I imagine that if it is their fault and they screwed up, "you're fired" is the appropriate thing to say, and generally pretty instantly.
The impact comes not from what they tell you, it's more about what they'll tell the next company when they call to ask about you (though if the two don't match up, you run into a lot of problems).
If the company has documentation to back up their side - something like "he/she was found asleep at his desk several times (dates A, B, C), was given both verbal (on date D) and written (on date E) reprimands, continued to sleep at his/her desk (dates F and G), and was let go on date H" - then I don't believe that there's a law that stops them from doing so.
Yes, they may get sued - but they're likely to win the lawsuit (especially if they required the employee to sign the written warning, so he/she can't claim ignorance).