A contractor is suggesting that 3- 4x4 posts can be placed on my timbertech decking to support a 12'x 20' wide gable roof. The 20' is where the posts would support. Then, under my deck, which is 6' off the ground, he would build supporting beam and new footings. Does that make sense to put 4x4's on top of composite decking?

No, it doesn't, unless there will be solid blocking from the underside of the synthetic decking to the top of the new beam.

Thank you. This makes more sense to me.