They're both grammatically correct.
One difference to me, as a native speaker, is that "He just fired his assistant" sounds less casual and more serious/respectful than "He's just fired his assistant."
The first one is more natural in a typical casual or professional conversation.
If you're speaking from a position of authority or judgement over the man, then "He's just fired his assistant" would sound natural as well.
If you're speaking of the man in a casual way, with a hint of condescension or head-shaking at his behavior, as if you've had to deal with multiple strange incidents from him, in the sense of "and now he's just fired his assistant too" then "He's just fired his assistant" would sound natural in that case.
"He has just fired his assistant" without the contraction, is different from both of the previous sentences. It's not the most natural choice for a casual conversation, but it is formal, and can be natural in a formal conversation.
Source: My personal experience.