Was the Prophet's fasting of Ramadan based on a fixed calendar, not on the sighting of the crescent moon?

The month of Ramadan was observed by sighting the crescent moon, and it rotated through the seasons of the year, unlike the Gregorian months. The texts of the Qur'an and the Sunnah confirm that months and years are known by the moon's movement and its phases, and that a year consists of twelve lunar months, and that the beginning and end of a month depend on sighting the crescent moon. This causes the lunar month to move between the seasons of the year. Incidents from the Prophet's biography confirm this; the Battle of Badr Al-Kubra took place in Ramadan, corresponding to March 13th, and the Conquest of Mecca, six years later, was in Ramadan, corresponding to January 10th, which indicates Ramadan's movement between seasons. The claim made by some researchers that the Arabs before Islam delayed Hajj has no valid basis.