Data, being a pedantic sort, as artificial lifeforms often are, has actually answered Picard's question literally.
PICARD :
A missile complex? ...
The date?
Mister Data, I need to know the exact date .
Note that Picard asks for the date and not the stardate .
Data is simply answering the question with the information that he thinks Picard wants according to his internal logic/programming.
In addition it has been mentioned before that Data thinks at a much faster rate
so he has probably had time to come to the conclusion that they are in the past and therefore that it will be more useful for him to give an old style date.
Data could instead have explained, at length, that the date was before stardates began but probably thought that it was unnecessary.
Wether or
not it's an in joke, deliberate plot point, or simply good/bad script writing is open for debate.
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This is perhaps circular, or at best, descriptive rather than explanatory, as it uses your very question as its reference point, but from Memory Alpha : Stardates ... do not apply retroactively instead of Gregorian or Julian calendars either: for example, the first contact with Vulcans still took place on April 5, 2063, not on a stardate.
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In-universe, TOS/TNG versions of Stardates can't be applied that far in the past , and they were the only two versions of Stardates that existed at the time that movie was created.
TNG/DS9/VOY-style Stardate 0 is the year 2323
TOS-style Stardates ran from 1312.4 through 4731.3, so Stardate 0 can be roughly estimated to be somewhere around 2263
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Four possibilities occur to me, which haven't been said yet...
Local context
I think the simplest answer is that when you're dealing with time travel and your Captain asks for the "exact date", you give it to him in the context in which it makes the most sense.
In this case, the Earth date, since they've traveled to a pre-stardate period.
Just like if we'd traveled to ancient Rome and you asked me "when is this," I might say, "the reign of Augustus Caesar."
I'd be using local context to describe the time period.
Stardates may be more useful for ships than planets
According to the Star Trek Guide ...
We invented "Stardate" to avoid continually mentioning Star Trek's century (actually, about two hundred years from now), and getting into arguments about whether this or that would have developed by then.
Pick any combination of four numbers plus a percentage point, use it as your story's stardate.
For example, 1313.5 is twelve o'clock noon of one day and 1314.5 would be noon of the next day.
Each percentage point is roughly equivalent to one-tenth of one day.
The progression of stardates in your script should remain constant but don't worry about whether or not there is a progression from other scripts.
Stardates are a mathematical formula which varies depending on location in the galaxy, velocity of travel, and other factors , can vary widely from episode to episode.
According to that description, stardates sound like a naval tradition, but not necessarily one that would be associated with planets.
On Earth you have no "velocity of travel" aside from the planet's natural movement, and your location in the galaxy is a given.
Combined with their presence in the 21st century before the "stardate era" began, it might render the formula irrelevant, or even incomputable.
At the very least it probably renders the idea of a stardate less useful and important, since those other variables aren't in play.
For example, a NASA scientist might say something is going "483 meters per second," but a cop on the highway would just say, "he's doing 95."
In the latter example, the unit of measurement isn't stated because it's understood in context.
Stardates might similarly be more useful for ships in transit than on planets, or on Earth specifically.
Stardates require computation Even if stardates are seen in the canon being used on Earth (and I'm sure they are), it might be more useful to give the Gregorian date in a pinch: as stated in the above quote, stardates are the result of a complex formula.
Even if Picard is capable of back-tracking a negative stardate into a Gregorian date equivalent to the era in which they've arrived, Data may just be saving time by doing the calculations for him.
The information source could be local When first estimating their arrival date, Data mentions taking measurements of the atmosphere, etc.
However, when Picard demands to know the exact date, Data doesn't elaborate on his methods for deducing it.
Given that the previous "natural scans" were accurate but imprecise, he could have accessed a computer on Earth or on a satellite, and pinged it for the current timestamp.
If this was so, it would likely have returned either a Unix timestamp, or a date/time string.
Either way, the most natural and efficient way to relay that information would be in the parlance of the time, which would be the Gregorian calendar.
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To expand on @Nerrolken's first point, note that Picard explicitly asks for the date they went to, not the distance they were sent back in time.
This suggests that his concern isn't getting back, or judging the capacity of Borg time travel - he's worried that there's something
specific they've been sent back to .
In that case, he needs a date that he can easily compare to what he remembers of history.
As for why he would learn history using Gregorian dates, there are two important reasons.
One, as @Izkata points out, is that stardates are troublesome that far out of their epoch.
More importantly, the prevalence of primary sources (Riker quotes directly from Cochrane, and Kirk actually met the man) that would naturally not use stardates would make it confusing to apply them to history.
So, if the Federation typically studies pre-Federation Earth history in Gregorian dates, and if Picard suspects that there's a historical event in progress that the Borg are going to interfere with, the Gregorian date is far more valuable in the moment than a stardate.
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In the movie Star Trek: First Contact the Enterprise enters the past through a temporal vortex created by the Borg. Upon arriving there is the following exchange:   PICARD : A missile complex? ...The date? Mister Data, I need to know the exact date.  DATA : April fourth, two thousand sixty-three.   Surely when the Earth converted to using  stardates  they converted all past dates into  stardates  as well. It isn't as if we refer to things in the  Olympiad  dating system.
Why didn't Data give Picard the stardate instead of the Gregorian date?
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Data is simply answering the question with the information that he thinks Picard wants according to his internal logic/programming. Stardates involve a complicated mathematical formula that requires computation. Even if stardates are used on Earth, it might be more useful to use the Gregorian date. Similarly, stardates might be more useful for ships in transit than on planets, or on Earth specifically. Data is simply answering the question without overthinking complex dating systems. Stardates could be more useful for ships than planets.  
Data is simply answering the question without overthinking complex dating systems. Stardates could be more useful for ships than planets.  