The Lunar Islamic calendar is based on a strict succession of months, with each month based only on the position of the moon, and no intercalated months or months extended to 31 days in order to keep the cycle of months aligned with the solar year. The start of the months are determined by the moon, the names of the months are not defined by any astronomical feature.
So to determine which Islamic month it is you need to know the actual amount of time that has passed. Now the positions of the planets change day by day, and by careful observation of the planets, and given a detailed ephemeris of their positions, one could work out the exact date, and therefore the Islamic month.
This is similar to determining the day of the week by astronomical observations, since the start of the day is based on the motion of the sun, but the name of the day is not defined on any astronomical feature.