Four possibilities occur to me, which haven't been said yet...
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.