Which one is the correct use?
She judged him wrong.
She judged him wrongly.
Or, are both correct, but have slightly different meanings?
This means she decided he was wrong in something he did or said.
This means the error was hers.
Usually you would express that this way:
But wrongly works as well.
There are two different structures. "Wrong" may be either the complement of the verb (what she judged him to be) or an adverb modifying the verb (how she did the judging).
As others have said, "wrongly" can only be the adverb, but "wrong" can be either - though in more formal use, it would tend to be used only as the complement.