Honesty,
I really don't know what veils have to do with being a muslim. This is a cultural thing, rather than a religious one. There are many translations of the Koran online which are slanted to fit some fundamentalist male's current view of what is permissible. There are two specific instances (24:31 and 33:59) that I can think of where women's clothes are discussed. First, I'll give any old online translation, and then my comments.
24:31
"And tell the believing women to reduce [some] of their vision and guard their private parts and not expose their adornment except that which [necessarily] appears thereof and to wrap [a portion of] their headcovers over their chests and not expose their adornment except to their husbands, their fathers, their husbands' fathers, their sons, their husbands' sons, their brothers, their brothers' sons, their sisters' sons, their women, that which their right hands possess, or those male attendants having no physical desire, or children who are not yet aware of the private aspects of women. And let them not stamp their feet to make known what they conceal of their adornment. And turn to Allah in repentance, all of you, O believers, that you might succeed."
Admittedly, this is a really poor translation from the word go, but please concentrate on the part that I've highlighted "
Illā Mā Žahara Minhā" and which I would argue (in the company of many scholars dating back centuries) does not mean "necessary" but is akin to "customary". Indeed, the whole point of this injunction is to get women to cover their breasts, for goodness sake. (Most men I know are more interested in breasts than hair.)
33:59 "
O Prophet! Tell your wives and your daughters and the women of the believers to draw their cloaks (veils) all over their bodies (i.e.screen themselves completely except the eyes or one eye to see the way). That will be better, that they should be known (as free respectable women) so as not to be annoyed. And Allah is Ever Oft-Forgiving, Most Merciful."
First, the entire chapter is heavily slanted towards the treatment of, and expected behaviour from,
the Prophet's wives. Second, the "veil" and "i.e." bits are again off. It is definitely "outer garments". The "i.e." bit is commentary provided by the translator.
Let me show you what, in my considered opinion, is dress forbidden to modest women in Islam by Ch.24:
and how women are not allowed to exit their houses in Ch.33:
(Incidentally, pretty much the same applies to men.)
Now, the Arabs having a long tradition of wrapping their women up; the entire Middle East does. It has been argued that this is originally a Sumerian tradition dating from the time of temple prostitutes, who were required to cover themselves. The greater the covering => the holier the woman. Women in Christianity "take the veil" and become nuns. It is equally a choice in Islam; there is no obligation to do it. If you have lived your life in a nudist colony, and you decide to become a muslim, you put on a dress. You don't need to wear a tent.
(I write as a dabbler in the Hanafi school of course, not one of those post-18th century innovators in Arabia.)
To return to what made me go off on this tangent the original comment
To those of you getting caught up on the non-religions aspects of the dress, think of it like this: Not everyone who wore turbans or veils was a Muslim, but virtually every Muslim of the period and place wore turbans, veils, etc. Is there a way to assign priority to certain religious aesthetics over cultural ones? If so, that might be a rather easy way to solve this.
Non-Middle-Eastern muslim women did not normally wear veils. Turkish women certainly didn't normally do so until the 15th or even 16th century, about 600 years after their embrace of Islam became widespread. As Arab influence spread, it became customary to do so. It is a cultural practice, not a religious requirement.
Sorry about the rant, but I feel strongly about this.
PS: If you think the Sumerian thing is stretching it, don't. The Sumerian god Dumuz(x) was adopted by the Babylonians, who gave the name "Month of Dumuz" to one of the months in their calendar. The deity's name is now rendered Tammuz. Which is incidentally the name of July in both the Hebrew calendar and in most of "extended" Mesopotamia, plus, with a vowel change, Turkey. Interestingly, four months prior comes the month of Nisan, which is the starting month of the Babylonian calendar and I believe (not sure about this) the religious Jewish one. (And again, used elsewhere too.) The Middle East has a lot of history and some things get passed on, without anyone really thinking about the origins of things any more.