In a word, yes, but only interpreting "western" broadly. The tradition of precise and systematic astronomical observations comes from ancient Babylonians, i.e. from Mesopotamia, and even Alexandria, where many scientists worked during Hellenistic times, is in the middle east. John Philoponus (c. 490 – 570) working there wrote things like "but this [view of Aristotle] is completely erroneous, and our view may be completely corroborated by actual observation more effectively than by any sort of verbal argument". One of the first practitioners of controlled experiments and field studies were an Arab polymath Alhazen (c. 965 –1040), and Persians Al-Biruni (973 – 1048) and Avicenna (c. 980 –1037). Ancient Babylon, Hellenistic Alexandria, and medieval middle east deeply influenced the scientific method tradition. They are often counted as "western" because they culminated in a completed form of scientific method, explicitly outlined by Francis Bacon (1620), in western Europe during and after the Renaissance, and then exported all over the world during the colonial period.
There was a remarkable period of scientific and technological innovation in China during Song dynasty (960–1279), that in some ways compares to Hellenistic period and Renaissance. The invention of gunpowder and nautical compass, as well as widespread use of printing press and paper money in China date to this period. Chinese Leonardo, Shen Kuo (1031–1095), the inventor of compass, also measured the distance between the North star and the true north, improved designs of the armillary sphere, gnomon, sighting tube, and water clock, devised a geological hypothesis for land formation, and proposed a hypothesis of gradual climate change. And yet Song China was a near miss, it did not lead to development of systematic theoretical and empirical science as we understand it today. But it shows that much can be accomplished in other traditions. A very sympathetic, anti-eurocentric, account is given by Morris in Why the West Rules - For Now, for balance here is a critical review of his book. But even Morris acknowledges that the East did not formulate a scientific method of its own.
Ancient Egypt and China, medieval India, American Indian civilizations, etc. also made many technological advances, but they were practically oriented traditional societies that developed their tools and procedures through accumulation of incremental glacial changes, generational trial and error, rather than through intentional innovation guided by scientific inquiry. Traditional medicine, aboriginal one in particular, is a typical product of such development, but so is ancient Babylonian or medieval Chinese mathematics.
Here is Stein's characterization of Babylonian mathematics, that applies to science in traditional societies more broadly:
"...for the Babylonians, mathematics was not an enterprise, but a lore and a skill possessed by the priestly or scribal class, for use in essentially administrative functions, and passed on from generation to generation much as were the techniques of the craft
guilds in the middle ages. Of course, the techniques had to have been
discovered; and innovations might occur from time to time; but
invention or discovery was no more the business of those trained in the
lore, than it was the business of the medieval master of a craft."