Is the woman who does not wear the Hijaab, in spite of acknowledging its obligation, considered a Faasiqah (of bad moral character)? What is the proof for that?

All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.

A woman who does not wear the Hijab, in spite of acknowledging its obligation, is a Faasiqah because she does not comply with the Commands of Allaah Almighty. Scholars said that Fisq literally means to get out of something. Ibn Jareer said, in his interpretation of the words of Allaah Almighty (which means): {…because they were defiantly disobeying} [Quran 2: 59], "It means, because they neglect the obedience of Allaah Almighty and they leave it for disobedience."

Shaykh Muhammad Yoosuf Al-Kafi At-Toonusi said in his book Al-Masa'il Al-Kafiyah:

Beauty is either natural or acquired.  A woman's face, neck, wrist, and so on are aspects of her natural beauty. Meanwhile, all that which is used by a woman to beautify herself is considered acquired beauty. Examples of acquired beauty are kohl, which is applied to the eyes, a necklace worn on the neck, a ring on the finger, a bracelet on the wrist, an anklet on the ankle, and colored clothes on the body. There are details regarding the ruling on a woman who reveals any part of her beauty, whether it is acquired or natural. That is to say, if she exposes something of what is stated above, believing that it is not permissible, then she has committed Fisq, and she has to repent of that. If she does that, believing that it is permissible, then she has committed disbelief because of her opposition to the Quran, as the Quran forbids a woman from exposing any part of her beauty to anyone except for those excepted by the Quran.  

Allaah Knows best.