In the canon about head covering for women, there are also prescriptions about women dressing modestly and men removing their hats. In our current Code (of Canon Law), this canon is omitted. However, if we take the stance that this omission means the Church no longer endorses it, than we also would have to say that the Church no longer desires women to dress modestly or men to remove their hats. I doubt any of us believes that. I cannot speak for those who worked on the current Code, but I imagine one of the reasons this canon was omitted was the hope that women would cover their heads and dress modestly and men would remove their hats without being told or forced to do so.
We need to make a distinction between what is acceptable and what is proper. It is acceptable in our society for men to wear hats indoors, but I do not think anyone would agree that it is proper or good manners. It is certainly acceptable for women to enter church with an uncovered head. However, it is a different question whether it is proper.
One poster named wearing a veil a devotion. That is fitting enough. However, the term devotion cannot be used to demean the custom. It is as much of a devotion as genuflecting on entering a pew, men not wearing hats, etc. In other words, the fact that it is a devotion does not mean that it should not be encouraged. Another such ‘devotion’ is frequent reception of Communion. The Church only requires that we receive once a year. Does that mean the Church does not want you to receive at other times? Quite the contrary. Is it possible to go to heaven without the ‘devotion’ of frequent Communion? Strictly speaking, yes, but this devotion helps a lot. It is not strictly necessary but it is pretty close. I am not saying that wearing a veil is just as helpful as receiving Communion. My point is that we need devotions, and a thing should not be discarded simply because it is a devotion.
The group most known for wearing veils is women religious. Nuns are not little pride-demons flaunting a holier-than-thou air. It is helpful to examine this group when discussing inculturation. Nuns are thoroughly western (in culture) but are the most free from secular pressure. My suggestion is that we look to these women for insight into what the difference is between things that are culturally acceptable and culturally proper. I am not saying that everyone needs to be a nun, but I think the confusion we experience in secular society is more clear to them.