Fabric armor exists BTW.
http://www.freepatentsonline.com/6526862.html
"A lightweight, ballistic resistant fabric armor constructed of multiple layers of high performance fiber woven fabric arranged in a quasi-isotropic orientation. The fabric armor is used in ballistic resistant garments to cover and protect vital portions of the human body. Used in a garment, the fabric armor is of minimal areal density and bulk while providing flexibility, freedom of movement, ventilation, and an inconspicuous appearance. The fabric armor meets many different global ballistic standards (e.g., the ballistic performance requirements of the National Institute of Justice Standard), with the number of woven fabric layers determined by the level of protection desired."
Sounds exactly like something Batman would wear.
Sounds good to me.
You could also easily exaggerate the concept. Base it on that, but have new explanations for the fabric, that doesn't exist yet in real life.
Say the fabric has metallic properties in it, that strengthen the armour side of things, given Batman better protection. You could also explain why the suit is grey, because of the metallic material that's used.
Something like that would look cool? and it gives the unique movie style, while remaining faithful to the comics.