The covering of the face in public really isn't part of Abraham's culture. Sara herself didn't cover her face or wear a marriage covering. That is why Pharoah and Abimelech sought her until it was revealed to them she was married to a prophet. They saw her beauty. Since she did not cover and there was no penalty, it stands to reason there was no actual law in Egypt or Caanan she could be punished by. They didn't beat her, beat her husband, or cut someone's head off. She was also in a foreign land, if breaking laws they would have punished her. The kings were more cordial and had more respect than modern Islamic leaders, they maintained dignity. Genesis 20 reveals Abraham provided her protection. He was her covering. Men wanting a relationship with Sara were given infirmities to prove he was a prophet and she was his wife. Abraham was the covering, not cloth. Its pretty clear she didn't cover her beauty or wear a marriage covering, since they didn't know she was married at first.
16 And unto Sarah he said, Behold, I have given thy brother a thousand pieces of silver: behold, he is to thee a covering of the eyes (mens eyes), unto all that are with thee, and with all other: thus she was reproved.
In the Quran the screen covering was only for the prophet's wives, not all believers. It was when eating in the prophets house. The prophets were given special rules so they could take wives they wanted, and they had protection from men taking theirs.
33:53
and when you ask his wife for something, ask from behind a screen; that is purer for your hearts and their hearts. You must never offend the messenger of God, nor must you ever marry his wives after him, for that would be an enormity with God.
John 7:23
If a man on the sabbath day receive circumcision, that the law of Moses should not be broken; are ye angry at me, because I have made a man every whit whole on the sabbath day?
The liability was on the man for looking and coveting. Women do instigate covetousness, but God made the woman's beauty with complete foreknowledge. Like the stars of heaven, God doesn't remove them even though men worship them. Men worship money but God doesn't take that away either. It is man's exercise to realize when something is wrong. We can learn to talk to opposite sexes without coveting.
Hebrews 5:14
But strong meat belongeth to them that are of full age, [even] those who by reason of use have their senses exercised to discern both good and evil.
In life we grow and learn what is right. Not all sin is avoided simply by custom.
If Mohammed were a prophet equal to Abraham his wives wouldn't need coverings. Mohammed would be their covering. We see a lot of hypocrisy in the Muslim world over coverings. I don't see them staying inside at night to avoid seeing the moon and stars, nor giving up money and credit cards to avoid serving mammon. Nor did Jesus remove the ability to lust after a woman, though he warned against it. Jesus could have imposed facial coverings but didn't.
God did not reveal to Abraham that his wife needed to cover her face or her body per se, Abraham was a prophet and God could reveal it. He didn't. Covering the hair was unnecessary because the hair is a natural covering created by God. Long hair covers the back and shoulders.
The covering in this case was the marriage ornament not facial covering. Abraham was her covering. The covering of the eyes wasn't for women, but for men who might look upon Sara and covet her. Those realizing God had consequences for coveting her would move on for another beauty.
You were not supposed to look upon or lust after a married woman and Abraham was her protection. God saved her for a miracle birth, conceiving at a late age. Men often covet other men's wives. The vail was a symbol like a wedding ring, people could know she was married. Those who approached Sara were stricken with calamity, which proved to Pharoah and Abimelech that Abraham was a prophet. Sara was reproved perhaps because she didn't wear a marriage covering, she had good reason because of the danger Abraham would be killed and she taken. I believe to protect Abraham in a country he was sojourning. It was dangerous to have a beautiful wife.
Sara made a wrong decision but by God's design, since God used the infirmities it brought to prove Abraham to some kings. It solidified his connection to God and because of that he was allowed the land.
There are many other evidences from Abraham's family that the women did not normally cover their faces and in many cases did not wear a marriage covering. Rebekkah travelled with men without a vail before meeting Isaac her husband. Once in his presence she wore a marriage vail on her head, not covering the face, there is no evidence for that. Some may think her a prophetess that may prophesy in her husbands presence, so a vail would be appropriate in his presence. Others see it as hiding the face to avoid being seen by the groom before the wedding.
Genesis 24
65 For she had said unto the servant, What man is this that walketh in the field to meet us? And the servant had said, It is my master: therefore she took a vail, and covered herself.
She had come from Abram's brother and knew the customs, she didn't practice Islamic face covering in front of men not her husband to be.
The marriage covering according to Paul was a symbol of authority for married women that was on the head, not the face. It likely wasn't a scarf. The point is she travelled without the vail. She wore the vail while taken to her mother in laws tent, by her husband. Why would she hide her beauty from her husband? The muslim belief that hiding beauty from temptation as the goal, the theory doesn't work well in the story, it falls apart.
The marriage vail was on the head, not the face. We see this in many Catholic churches with small beaded lace coverings that don't hide beauty.
10 For this cause ought the woman to have power on her head because of the angels.
This is likely from Genesis 6 where it says the women were "taken", implying the married as well. The word taken implies stolen or removed by instigation. The people who belonged to God corrupted themselves and marriage, by taking other men's wives for their own.
2 That the sons of God saw the daughters of men that they were fair; and they took them wives of all which they chose.
The symbol of authority spoken of by Paul, or marriage vail was a symbol to all men and women after the flood, a symbol that marriage was sacred. It should remind muslims as well, who practice taking the spoil of non-Muslim women, that God doesn't approve of taking what you choose. In Genesis 6 wearing vails wasn't practiced but was later known in Abraham's day. It wasn't face covering. Genesis 6 shows the women were open to see their beauty.
The flood of Noah it seems came partially due to desecration of marriage, so it would be stressed after Noah. Coverings were adopted for the married, a custom, but it wasn't law. God could make all women be hidden for eternity as some sects of Islam have or simply create a symbol to remind mankind of the cause of Noah's flood. A symbol showing a woman is married and it is improper for another man to take her. Men should consider the consequences and that is a deterrence.
Those who covered the face did so to hide their identity, not to hide beauty. Face coverings protected evil doers from being identified. Harlots, taken wives being transported, those engaged in adultery, spies, etc had reason to cover their faces. It should be a woman's choice. There are reasons to cover the face. Sand storms, bright sun that can burn the face, there are reasons. Yet, it seems it became law after Islam started in the heighth of the slave trade. 1 million women were carried from Spain and Portugal from Islamic invasions, taken to African and Arab countries. It was just Islamic slavery.
Genesis 38
15 When Judah saw her, he thought her to be an harlot; because she had covered her face.
19 And she arose, and went away, and laid by her vail from her, and put on the garments of her widowhood.
The harlots tended to cover themselves, and laying aside the vail after deceiving Judah showed a widow of sexual age did not wear a vail publicly. She took it off once done deceiving.
The point is, Abraham's family did not wear face coverings, and even the vail wasn't law. Symbols of marriage are ok, but Islam corrupted the purpose. It wasn't to hide women. Men who have been married and know a woman's nakedness can fantasize and imagine women no matter what they wear. Skimpy clothes make it worse, but you can't stop lust by force. Muslims have as much lust or more. They aren't holier than any others.