Carnatic music is culture specific in much the same way as all arts are.
However, all great arts ultimately transcend all barriers and can powerfully affect the human consciousness at large. One doesn't need to be a 'Christian' to appreciate Da Vinci's 'The Last Supper or a 'Hindu' to appreciate Saint Thyagaraja's music. Art exists in a realm of its own where the Dikshitars, Shakespeares, Bachs, Kafkas, Coltranes and Van Goghs happily live. Its my belief that Carnatic music, much like Western classical music and Jazz is so comprehensive and highly evolved a musical tradition that it is universal in its scope and application. An insightful observation of Dikshitar's masterpieces can be an object lesson to symphonic composers on how to develop long forms and elaborate structures, building powerful climaxes at the right points, etc.