Although it is difficult to claim in which civilization clothes were invented. If we go by evidences, Cotton was cultivated in India since 5000 BC. Inhabitants of Mehrgarh wore fine clothes. Many experts agree that primitive gins and spinning wheels originated from India. The earliest samples of cotton fabric were found in the excavations of Harappa. A lot might clear up if we can understand their script.
Samples of the most ancient mordant dyeing technique for cotton fabrics were also found in remains of Harappan civillazition. Ancient texts describe spinning and weaving and the geometries of their construction, which themselves are interesting concepts to read. Indians used fibers for spinning which were of many types, unparalleled in any other civilization of the period.
India exported large quantities of sheep wool, woolen fabrics, colored carpets, silks, cotton clothes, and fabrics to other countries, including Rome, Greece, Egypt and Mesopotamia. How ironic it is, that today Indian cotton farmers are committing suicides.