I think veiling is an extreme form of ownership, but ownership is ownership, no matter how tacit the laws. In America, we place a gold ring on the left finger and we expect society to respect that ring every bit as much as Islam expects society to respect the veil. Whether in respect to one institution or another, whether for one reason or another, it is all about ownership. Scripturally, God lays claim to ownership of our souls, we belong to Him in jewish, christian and muslim texts. Catholic robes for priests and half-veils for nuns express that ownership. And again, that ownership stems from marriage; God has married us, taken us as his bride.

What would American society say about a married woman who refuses to wear her wedding band?

The veil hides the human face and so it becomes a type of repression, but it wasn't designed solely for males to exert sexual dominance, in fact, I'd say that was secondary to the primary desire to mark out souls in ownership... something that all societies do, albeit with different tools, methods, and reasons. Repression results from the ownership more than any other factor. In America, a married person is free to express themselves to the opposite sex, but they are not free to cross over the boundaries or ownership... they are not free to "cheat" on their spouses.

I really think ownership is the central issue and reason for veiling as well as other like social practices.