[Petronski]

A muslim person would follow islamic law. But would it be less accurate to say a muslim person would follow muslim law? Or perhaps just less elegant?

I don't know, nor frankly am I all that concerned with getting it right. But it is intellectually interesting.

[highlander_UW]

A muslim person would follow islamic law. But would it be less accurate to say a muslim person would follow muslim law? Or perhaps just less elegant?

I'm not a specialist in grammar, but my understanding is that the first would be correct, a muslim would follow islamic law. The source of the "law" would be Islam, not the muslim himself...so to say muslim law would be inaccurate. (although most would understand the intended meaning).

[Asphalt]

The latter.