King of the Dead vs Witch King

[Boromir88]

the Dead King is not a living man.

But the Dead King has no body, so he could not literally inflict damage on him (as in punch, stab, or kill).

Also, the WK is not above fear, so there is a chance the Dead army could have made him retreat

No, hte WK, is not above fear, but why would the Witch-King be afraid of the very fear that he uses and the rest of the Nazgul:

Their peril is almost entirely due to the unreasoning fear which they inspire (like ghosts). They have no great physical power against the fearless.~Letter 210

I would call the Dead Army's fear, a 'ghost-like' fear, and the way the Corsairs were sent into a maddening jump to their death. So, why would the WK fear the very fear that he inspires? Or vice versa?