Strictly speaking, no.
In fact, strictly speaking, a Fighter 16/Justice of Weald and Woe 2 could not take Weapon Specialization as his 18th-level feat. The rules text in question:
Use your justice level when determining whether you meet a prerequisite of a minimum fighter level.
Not “you may” or “you can,” but a directive. And nothing about the wording suggests that it’s only for the purposes of the bonus feats that you get. Thus, he would count only as a Fighter 2 for the purpose of the prerequisite. In fact, he would lose the ability to use Weapon Specialization if he already had it, because you must meet a feat’s requirements to use it.
This is, pretty obviously, dumb.
Typically (as in, in just about every case I can think of where something allows you to qualify as a Fighter), the rule specifically says that they stack. I’m reasonably confident that the Justice of Weald and Woe is not intended to be an exception to this, but rather the lack of that term was an oversight.
All that said, there is only one feat that requires Fighter levels that is any good (Weapon Supremacy), and considering how many other (awful) feats you have to take to qualify for that one, you are probably better off not qualifying for those feats. It prevents you from falling in the trap of taking them.