This thread was different from what I expected. It seemed to be focused on "should I veil when I pass a church" (short answer: no).
I thought it was a thread about "veiling" or the practice in some cultures of wearing a veil basically 24/7, like the Muslims and other groups.
If that was the topic, I was going to say that
A) there's nothing wrong with it
B) it is consonant with promoting proper gender roles, and certainly doesn't go against them
C) but it's not part of our culture. Culture isn't about right and wrong or sin.
Who is right, the Italians who prefer wine, or the Germans who prefer beer? Both and neither.
So I would advise American women NOT to practice "veiling" (defined as wearing a veil at all times) because you're going to be taken for a Muslim, because it's not part of the culture in America, not even in a certain group (Catholics or Traditional Catholics) in America.