Short answer: Yes, Islam does not give a choice to women. They are commanded to cover themselves up. Moreover, Muslim men are given authority to enforce this.
The Scripture of Muslims (Qur’an):
Quran 24:31–“And tell the believing women to lower their gaze (from looking at forbidden things), and protect their private parts (from illegal sexual acts) and not to show off their adornment except only that which is apparent (like both eyes for necessity to see the way, or outer palms of hands or one eye or dress like veil, gloves, headcover, apron), and to draw their veils all over Juyoobihinna (i.e. their bodies, faces, necks and bosoms) and not to reveal their adornment except to their husbands, or their fathers, or their husband’s fathers, or their sons, or their husband’s sons, or their brothers or their brother’s sons, or their sister’s sons, or their (Muslim) women (i.e. their sisters in Islam), or the (female) slaves whom their right hands possess, or old male servants who lack vigour, or small children who have no sense of feminine sex. And let them not stamp their feet so as to reveal what they hide of their adornment. And all of you beg Allaah to forgive you all, O believers, that you may be successful”
Quran 24:60–“And as for women past childbearing who do not expect wedlock, it is no sin on them if they discard their (outer) clothing in such a way as not to show their adornment. But to refrain (i.e. not to discard their outer clothing) is better for them. And Allaah is All‑Hearer, All‑Knower”
Note: ‘Women past childbearing’ refers to those women who are too old to be pregnant.
Quran 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 Allaah is Ever Oft‑Forgiving, Most Merciful”
Quran 33:53–““O you who believe! Enter not the Prophet’s houses, unless permission is given to you for a meal, (and then) not (so early as) to wait for its preparation. But when you are invited, enter, and when you have taken your meal, disperse without sitting for a talk. Verily, such (behaviour) annoys the Prophet, and he is shy of (asking) you (to go); but Allaah is not shy of (telling you) the truth. And when you ask (his wives) for anything you want, ask them from behind a screen, that is purer for your hearts and for their hearts. And it is not (right) for you that you should annoy Allaah’s Messenger, nor that you should ever marry his wives after him (his death). Verily, with Allaah that shall be an enormity”
The Traditions of Muhammad (Hadith):
Sahih al-Bukhari 4758–Narrated `Aishah: May Allah bestow His Mercy on the early emigrant women. When Allah revealed: “… and to draw their veils all over their Juyubihinna (i.e., their bodies, faces, necks and bosoms)…” (V.24:31) they tore their Murat (woolen dresses or waist-binding clothes or aprons etc.) and covered their heads and faces with those torn Muruts.
Sahih al-Bukhari 6240–(the wife of the Prophet) `Umar bin Al-Khattab used to say to Allah’s Messenger (ﷺ) “Let your wives be veiled” But he did not do so. The wives of the Prophet (ﷺ) used to go out to answer the call of nature at night only at Al-Manasi.’ Once Sauda, the daughter of Zam`a went out and she was a tall woman. `Umar bin Al-Khattab saw her while he was in a gathering, and said, “I have recognized you, O Sauda!” He (`Umar) said so as he was anxious for some Divine orders regarding the veil (the veiling of women.) So Allah revealed the Verse of veiling. (Al-Hijab; a complete body cover excluding the eyes).
Sahih al-Bukhari 5466–Narrated Anas: I know (about) the Hijab (the order of veiling of women) more than anybody else. Ubai bin Ka`b used to ask me about it. Allah’s Messenger (ﷺ) became the bridegroom of Zainab bint Jahsh whom he married at Medina. After the sun had risen high in the sky, the Prophet (ﷺ) invited the people to a meal. Allah’s Apostle remained sitting and some people remained sitting with him after the other guests had left. Then Allah’s Messenger (ﷺ) got up and went away, and I too, followed him till he reached the door of `Aisha’s room. Then he thought that the people must have left the place by then, so he returned and I also returned with him. Behold, the people were still sitting at their places. So he went back again for the second time, and I went along with him too. When we reached the door of `Aisha’s room, he returned and I also returned with him to see that the people had left. Thereupon the Prophet (ﷺ) hung a curtain between me and him and the Verse regarding the order for (veiling of women) Hijab was revealed.
Sahih al-Bukhari 372–Narrated `Aisha: Allah’s Messenger (ﷺ) used to offer the Fajr prayer and some believing women covered with their veiling sheets used to attend the Fajr prayer with him and then they would return to their homes unrecognized .
Sunan Abi Dawud 1833–Narrated Aisha, Ummul Mu’minin: Riders would pass us when we accompanied the Messenger of Allah (ﷺ) while we were in the sacred state (wearing ihram). When they came by us, one of us would let down her outer garment from her head over her face, and when they had passed on, we would uncover our faces.
Hijaab al-Mar’ah al-Muslimah, p. 54-67–The conditions of hijaab. Firstly, It should cover all the body apart from whatever has been exempted. Secondly, it should not be an adorment in and of itself. Thirdly, it should be thick and not transparent or “see-thru”. Fourthly, it should be loose, not tight so that it describes any part of the body. Fifthly, It should not be perfumed with bakhoor or fragrance.
Shaykh Muhammad Saalih al-Munajjid–At the time of the Prophet (blessings and peace of Allah be upon him) women were keen to cover themselves completely, which includes covering the feet. They used to let the hem of their garments hang low. One of them asked the Messenger of Allah (blessings and peace of Allah be upon him) about letting this hem touch what is on the ground, which may make it dirty, and he said: “That which comes after it purifies it.” On another occasion, when he (blessings and peace of Allah be upon him) said that women should let their hems down by one hand span, one of them said, “But then her feet will show.” This indicates that they were keen to cover their feet and that the feet are part of that which must be covered.
Group of Muftis, Islam Online–Fathers and mothers should note that they are responsible before Allah for the affairs of their daughters that have been entrusted to them; they should raise their daughters according to the Islamic manners. So if a girl is approaching puberty, there is the fear that her not wearing hijab may cause young men to be tempted by her or her by them. Hence in this situation her parent or guardian has to make her wear hijab so as to prevent means that may lead to evil or immorality.”
Former Muslim notes:
Islam commands that every Muslim woman should wear hijab and cover their faces and bodies. Those who do not obey this command are subjected to floggings (or other Islamic punishments).
According to Sharia, it is mandatory for Muslims to enforce hijab on all women. As Jamal Badawi (CAIR) said that hijab is “a command of Allah to Muslim women.” Therefore, it should be “the duty of the state” to establish it.
Moreover, Islam does not subscribe to the idea of ‘freedom of dress.’ Women are compelled to wear a burka even when they don’t want to wear it. Islam allows Muslim parents, brothers, and other elders to force women of their household to wear hijab.
The reason for covering up is to keep men from being tempted. As one Imam puts it, Sheikh Taj Sin al-Hilali said,”If I came across a rape crime – kidnap and violation of honour – I would discipline the man and order that the woman be arrested and jailed for life.”
“Why would you do this, Rafihi? He says because if she had not left the meat uncovered, the cat wouldn’t have snatched it… If you take uncovered meat and put it on the street, on the pavement, in a garden, in a park or in the backyard, without a cover and the cats eat it, is it the fault of the cat or the uncovered meat? The uncovered meat is the problem.”
This Imam is a mere reflection of the enormous problem in the Islamic religion.