There is no veil in Hinduism. Purdah or Pardha, Hijab etc were not part of Hinduism and it is not found in the Vedas, Dharmashastras, epics or other representative Hindu texts. The institutionalized form of veil, known as ghunghata (Ghunda) among Hindus (especially in North India) began with the arrival of Islamist invaders. The Rajput women and the Hindu women of Northern part of India used the veil to guard themselves from the predatory eyes of Muslim men.
The women in ancient Hinduism (Sanatana Dharma) had an equal status with man and freely participated in social and religious ceremonies without covering her face.
Women poet seers, female teachers, wives of acharyas, students of Bahvrca school, singers (Rig Veda IX. 66.8), dancers (Rig Veda I. 9.1.4), warriors (x.39-40) figure in Rig Veda without veil.
A marriage hymn solicits the guests to see the fortunate bride (Rig Veda X. 85.33).
Vadhrimati and Vispala fought in the battle field (Rig Veda 26.40) without wearing veil.
Mudgalini drove her husband’s chariot (Rig Veda 102.2-3) and Gargi challenged Yajnavalkya to debate in the assembly of King Janaka of Videha (Brihadaranyaka Upanishad 3.6, 3.8) without wearing a veil.
The grammarian Panini refers (Ashtadhyayi 4.1.49) to female students as chatri and their hostels as chatrisala, but nowhere does he mention about the veil.
Females in Ramayana and Mahabharata did not wear any veil. The prevalence of swayamvara form of marriage in which a girl chooses her husband from among several suitors shows that women were not kept sequestered in ancient Hindu world.
Even Smriti literature does not mention the use of veil.
Ancient Hindu sculptures do not show females behind veil. All goddesses, saptamatrikas, mahavidyas, amman forms, village goddesses appear without any kind of veil.
The Chinese travelers Fa Hien Hiuen tsang and I Tsing do not mention the use of veil. Abu Zaid, an Arab writer of the early 10th century AD, noted that women could be seen in royal courts without veil.
Firuz Shah, the second sultan of the Tughlaq dynasty (1351 – 88 AD) and brutal Muslim ruler enforced purdah on Hindus in accordance with Islamic injunctions. The Hindu rulers and princes started enforcing purdah to save women of the royal houses from prying eyes of Muslim rulers and nobles.
The socio-religious reformers of the 19th century, like Raja Rammohan Roy (1772 – 1833), Swami Dayananda (1824 – 83), Swami Vivekananda (1863 – 1902) and others condemned the pernicious institution of purdah which had long enslaved women and impeded their progress.
Veil or purdah has been completely abandoned by Hindus. Its survival in the form of ghunghata is more a matter of social demeanor than a matter of religious concern.
Notes taken from Encyclopedia of Hinduism Volume VIII page 328 – 29 IHRF