Did she judge him "wrong" or "wrongly"?

Solution 1:

She judged him wrong.

This means she decided he was wrong in something he did or said.

She judged him wrongly.

This means the error was hers.

Usually you would express that this way:

She judged him unfairly.

But wrongly works as well.

Solution 2:

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.