Did ancient & medieval indians have a weekly day off? Or was there even a generally recognized day as public holiday? Or did the concept get introduced with Arabs or Europeans?
As one might expect, it is a tricky business to talk generally about practices on the Indian subcontinent. For most things that is as true today as it is for the subcontinent of centuries ago. The diversity of religions, cultures, languages, and the complex political realities over the centuries means that this answer really must be more of a sampling of the variety of practices.
Below I offer a sampling of what I have found on the practice of rest days in Indian Buddhism, Zoroastrianism, Jewish Indians, Hindu festivals, possibly regional rest days, and conclude with possible emergence of greater standardization with British colonial rule.
Uposatha and the Claim of Earliest Rest Days
The closest I came to something pointing to a very ancient origin is in an article by Bruce A. Kimball. I think this quote is also important because it points to the connection between feast days, taboos surrounding them, and rest days:
This connection, which traces the emergence of rest days in both the Buddhist uposatha (On Wikipedia) and Sabbath contexts again here in article by Edward Westermarck:
On when the Uposatha was held for Buddhists, with claim that it did not originate with Buddhists:
The Jewish population, especially on the west coast such as the Bene Israel, known as the Shanivar Telis (Saturday oil men?), keep to Saturday for their Sabbath [8:489]. The Zoroastrians in India have the interesting concept of intervals called Gahambars:
Hindu Festivals and Regional Diversity
Elsewhere, everything I find shows incredible diversity. Here and there, but not in sources that seem to be particularly well researched, there is talk of Saturday as a weekly holy day, and some claim this corresponds to the traditional "oil bath" in parts of southern India, which could take many hours - something you can read about in Ashtanga yoga as well. However, I didn't find much in the way of detail.
Instead you find a lot of passages like this from Caleb Wright in 1853:
Others note local practices like Abbe DuBois in Mysore in 1906:
There are less frequent Hindu festivals that include days of rest, including:
And also some limited to certain tribes:
I don't see much on the Muslim population. Globally speaking it looks like Jumu'ah on Friday (Wikipedia) , while a day of prayer and singled out, only really takes on its "rest day" in full form in some muslim countries in the process of modernization. I didn't find much on its application in pre-colonial India.
In the answer by @Kobunite we see the implementation of legally defined days of rest, but it looks like this process of Sundays being a customary day off starts to catch on during the course of the colonial period.
An interview of a worker in Bombay from late 19th century also speaks of Sunday as the standard:
The "first meeting of industrial workers in Bombay's history" in 1884 passed a resolution asking for a full day of rest on Sunday [10:328] and an "Act of 1891" made Sunday a day of rest in factories [11:230]
Overall, it would appear that in some areas or among some religious groups there are more regular festivals or religious days, with some cases of more frequent regular days of rest such as the observed weekly day of rest in Mysore, or the almost weekly Uposatha in some places like Sri Lanka, or the Gahambars of the Zoroastrians.
Sources are cited in format [Source Number:Page Number]
Well, as stated by Monster Truck there are many theories about when the "weekly off" was introduced to India. For those of you who are interested, one of the more common theories is that it was introduced during British rule in the country due to the observance of the sabbath. It is also highly likely that rest days were observed amongst members of the various religious groups in India. As a result, it is likely to be nearly impossible to accurately judge when the concept was first introduced to the sub-continent.
The Weekly Holidays Act applies specifically to Restaurants, Shops and Theaters and specifies a legal requirement for workers to be accorded a weekly day off.
The Minimum Wages act, obviously, defines the need for a minimum wage for people working in India but it also defines the right for workers to have a weekly day off.
As I said above, it is difficult to judge when the concept itself was introduced to India due to it's very nature. However, what I have said above shows the legal implementation of the concept.