Io (tarda) Saturnalia! (Hail (belated) Saturnalia!)

At first glance, Saturnalia looks like a rather debauched cousin of Christmas. Schools and businesses closed. Gifts were exchanged. Revelers filled the streets. These commonalities, however, are superficial. Christmas and Saturnalia have one, and only one, thing in common: they more or less indirectly mark the winter solstice. Their similar-looking traditions evolved independently.

There is a better case, however, for claiming that the date of Christmas was determined by a Roman festival. In late antiquity, a festival called the Nativity of the Unconquered Sun was celebrated on December 25 (the winter solstice in the Julian calendar). Though he became more prominent from the third century onward, the Sun (Sol) was not a particularly important god. Nor was his Nativity the most significant of his festivals; all we know about it is that it was celebrated - in the city of Rome, at least - with chariot races. But since it was held on Dec. 25, some scholars have theorized that the celebration of Christmas was moved to that date to overshadow (if you will) the Sun's festival.

The evidence is certainly suggestive. Christmas only became an important festival in the fourth century, at a time when the church was struggling to define its imperial mission. Several late antique bishops felt compelled to remind their congregations that Christmas commemorated Christ, not Sol. A few medieval scholars state outright that Christmas was moved to the 25th to outshine the Sun.

Modern historians, however, are less sure. An alternative theory suggests that the date of Christmas had nothing to do with Saturnalia, the Sun's Nativity, or even the winter solstice. As this theory points out, early Christians assumed that Jesus, as a perfect being, led a mathematically perfect life. By this logic, since the date of the Crucifixion was calculated to have been March 25, Jesus must have also been conceived on March 25. His birth would have occurred precisely nine months later, on December 25.

I made a short video about the Roman antecedents of Christmas, which provides a little more information about Saturnalia. This page has the video link and a short list of helpful sources.

You might also be interested in this old answer on Saturnalia gifts.

What follows is a response to a similar question several years ago by Archaic Greek specialist /u/rosemary85, who deleted his account and some answers a few years ago. I do not think it contains any information that would identify him in real life, so I thought it acceptable to put forward. His scholarship I thought impeccable and his answers always well-thought out. I did not put it into block quotations because that would nest some other quotes he put in. Apologies for any issues with formatting, which I may have messed up after pasting it.

(start response)

Even scholarly sources get tainted by inference on this topic, so perhaps it will be of some use to expose you to some of the primary evidence.

In the first place, some other posts I've seen here cite 354 CE as the year that Christmas was instituted. What that date actually represents is the earliest attestation of Christ's birth being celebrated on 25 December, in a document known as the "354 Calendar" or the "354 Chronology". Here's a copy of the relevant part of the text.*

But "earliest attestation" is not the same thing as "that's when it was first instituted". The likelihood is that that choice of date was earlier; though it's up for debate whether that means a couple of decades earlier, or a couple of centuries earlier.

(* Edit. Oops, linked to the wrong part of the 354 Calendar. Fixed now.)

Now, there are some earlier sources that could potentially be thought of as pointing towards celebration of Christ's birth -- or the Incarnation, depending on the theological flavour of the source -- at an earlier date. They are:

1. The late 2nd century/early 3rd century writer Clement of Alexandria at one point claims that some people assigned Christ's birth to "the 25th day of Pachon", and others to "the 24th or 25th day of Pharmouthi". These are dates in the Alexandrian calendar, which had been adopted as a standard solar calendar in parts of the eastern Roman Empire ca. 30 BCE. When you translate these dates into the Julian calendar they're nowhere near December (they come out as 20 May and 19/20 April, respectively), but the focus on the 25th (or 24th) day of the month is rather conspicuous, even so. Of course it could be coincidence. Moreover, there are alternative interpretations of Clement's dates: see S. Roll, Towards the Origins of Christmas (Kok Pharos: Kampen, 1995), pp. 78-9).

2. A reference to Christ's birth on 25 December in an early 3rd-century writer, Hippolytus of Rome's Commentary on Daniel 4.23. This reference is useless, though: it's definitely an interpolation of mediaeval date. One manuscript of Hippolytus omits the date altogether; another may possibly give the date as 2 April (but that text appears to be corrupt); and other writers of the first millennium quote Hippolytus differently. See further this open-access page; S. Hijmans, "Sol Invictus, the winter solstice, and the origins of Christmas", Mouseion 3.s. 3 (2003): 377-98, at 377 with n. 3; and Roll, pp. 79-81, especially p. 80 with n. 106. The upshot is that most textual critics accept the 2 April date as being, if not the original text, at least closer to what Hippolytus wrote than the 25 December date.

3. A statue of Hippolytus of Rome, dated to 222-235, has inscribed on it a table converting the date 14 Nisan in the Hebrew calendar (the traditional date of Pascha for early Christians outside Judaea) to the Julian calendar for various years. Since Pascha shifted around relative to gentile calendars, it was useful to have a reference table. For one year, ca. 30 CE, 14 Nisan is converted to 25 March, potentially suggesting a death-date for Christ on that day. A convoluted argument could feasibly maintain that this also represents the date of his conception; using a straightforward nine-month gestation period, this would put his birth on 25 December. Again this could potentially be coincidence. More details in Roll, pp. 79-80.

4. A text falsely attributed to Cyprian, the De Pascha computus ("computations concerning Pascha"), written in 243, at sections 18-23 puts the creation of the cosmos on 25 March; the creation of the sun on 28 March; and therefore also the genesis of Christ, the "Sun of righteousness" (a phrase in Malachi 4.2 which according to Christian tradition refers to Jesus), on 28 March. The author expressly claims divine inspiration for these dates. The word genesis would normally be translated "birth", but a number of writers use it instead to refer to conception -- for example, Clement of Alexandria at Stromateis 3.12.83.2. On that interpretation, that would translate to a birth-date of 25 December or 28 December (depending on which date you think pseudo-Cyprian made up).

All of these earlier references are at worst simply wrong, at best uncertain. So 354 ends up as the earliest uncontroversial date for an assignment of Christ's birth to 25 December.

As an endnote I should perhaps mention that there is basically no reason to suspect that Christmas was adapted from one of various pagan festivals. The usual candidates are the winter solstice, Saturnalia (17-19 December), Brumalia (exact date unknown), and a festival of Sol Invictus "the unconquered sun" (25 December).

1. *Winter solstice: 25 December is not the winter solstice, and the ancients were perfectly well aware of that fact. Moreover, there were no festivals dedicated to celebrating the winter solstice (until Brumalia came along, at a late date: see below).
2. [Edit, written a bit over a month later. It turns out this isn't quite so simple. Around the time of Jesus' birth, say 1 CE, the solstice did in fact fall on 25 December (Julian calendar). However, in 325, the date of the Council of Nicaea, at which the date of Easter was formally decided, the solstice fell on 19 December (Julian). Given that, and given that Christmas never had a shifting date to be in synch with the solstice, it'd be hard to sustain a case for Christmas as a solstice festival.]
3. * Saturnalia: also the wrong date; there's nothing to link the two festivals in terms of specific customs; and Saturnalia continued to be celebrated alongside Christmas after the Christianisation of the Roman Empire.
4. * Brumalia: in some ways an even worse candidate than Saturnalia, as it's not attested very early (earliest attestation is in the late 2nd century CE), and the date is even more wrong (it was initially celebrated on 24 November). It too continued to be celebrated alongside Christmas after the Christianisation of the Empire, at least as late as Justinian. On the other hand, in the Byzantine era it was combined with Saturnalia and some customs associated with Saturnalia/Brumalia (which included gift-giving, in the case of Brumalia) may possibly have been reassigned to Christmas as the pagan festivals gradually dwindled in importance.
5. * Sol Invictus: a festival for Sol Invictus is first attested in the 354 Calendar, the same document that gives us Christmas on 25 December. So there's nothing to suggest which had priority. More importantly, there's nothing to suggest any connection between them (other than the date): this festival of Sol Invictus (there were other festivals of Sol) was a state cult associated with a specific temple; Christmas wasn't localised in that way.

TL;DR: the earliest evidence for Christmas on 25 December comes from the year 354. It was most probably earlier, but how much earlier is a matter for dispute. There is nothing to suggest Christmas was an adaptation of an earlier pagan festival.

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So much for clearing away misinformation. Actually answering the question as posed is a lot harder, and I think needs a specialist in early Christianity. One old post on the subject by /u/Gadarn states the following, and it seems to be a widespread view though I'm not familiar with the evidence for it:

The most popular view is that early Christians derived the date from the date of the Passion of Christ which had been established earlier as March 25th. The idea is that the time between Christ's conception and death was exact to the year so, if he was conceived on March 25th then he was born on December 25th.

In another old post, /u/talondearg points out

The date of Christmas was calculated on the belief that great prophets and the like, were born and died on the same day of the month. The historian Tighe argues, and some patristic evidence supports, that when Easter was fixed, it lead to the calculation of Christmas as 25th Mar. The only rival date was Apr 6th, based on different calculations of Easter (yes, I realise Easter is movable because of the lunar calendar, but we are talking about the calculation of the first Easter in particular).

I don't have the chops to support or refute their view; only to say that this appears to be the mainstream view on the subject.

(end response)

If anyone has any issues with this due to its deletion, please let me know; while I respect his right to delete his responses, I thought it a shame to let this rundown of the sources be lost.