Fence gates themselves can be placed upon grass paths (grass block smoothed with a shovel) without un-paving the grass (turning to a normal dirt block) - however, if you place a fence gate on a grass block, you cannot turn the grass block underneath into a grass path no matter what angle you try from with a shovel.

What I've found is that, when a normal fence post (not a gate) is placed on a grass path block, it will immediately turn the path underneath to dirt. Fence gates will not turn a grass path into a dirt block, so I assume this post-placement paving issue comes from fence gates being conceptually similar to fence posts.