Firing people should [usually] be hard. Regardless of their work performance, a person's job is often their livelihood and when you terminate their employment, you're taking an active role in making their life harder.
That's not to say that you shouldn't fire someone for legitimate reasons. But if you do, you should try to be as fair and generous as possible. Whining about someone saying you "lack empathy" on LinkedIn as though you're some sort of victim definitely shows a lack of empathy.
I've had to fire a handful of people in my career. I felt terrible about all but one of them and tried to make sure they got the best severance possible.
The one I didn't feel bad about stole money from poor people and was completely remorseless. He can (and did) get fucked.