The short answer is no, because all Biblical dates that we can convert to modern calendars come via either astronomical observations that we can back-calculate OR by conversion to the *Julian calendar.
Two examples:
We have Babylonian astronomical observations paired with Babylonian calendar dates. The laws of motion allow us to calculate exactly how long ago a particular astronomical phenomenon occurred, thereby tying a Bablyonian date to a date on a modern calendar. This is how we know when Nebuchadnezzar's reign began and when a number of events during his reign occurred.
We have Roman officials whose terms of office were known, such as the proconsul Gallio before whom Paul appeared while in Corinth. We know when Gallio was in office (+/- about a year), in terms of the regnal years of Emperor Claudius, and we know on the Julian calendar when Claudius was in office. From which we can calculate the timing of many events in Paul's ministry.
Perhaps what you're looking for, though, is whether the fall of Jerusalem can be dated by something other than Babylonian dates. Possibly...but in ancient chronology the more certain dates are used to ascertain the less-certain dates, so we would get a less accurate approximation by trying to date the fall of Jerusalem by back calculating from something later, like known events from the Persian Empire.
But even then, we'd still have to tie the Biblical chronology back somewhere to an external record of an astronomical event, or the Julian calendar. The Bible doesn't provide direct links to the Gregorian calendar, we always use an intermediary to do that (whether it's the Babylonians or the Persians or the Greeks etc, there's no getting around that).
Even if we developed a thoroughly internally consistent Biblical calendar using only Biblical texts, and could thereby determine how many years apart two Biblical events were, without astronomy or secular history we couldn't figure out how many years there are between those events and today.
*We could also mention the Greek Olympiad dates or the chronology from Josephus, but these are known by conversion through the Julian calendar as well.
