It's very unlikely your stone is a fossil. Sandstones in particular often get concentric weathering rings, not an official term, where water migrates into the stone and discolours through subtle mineralization the stone. The concentric rings are also common on stones developed as concretions, again fairly common in sandstone. Given the texture of the stone from the picture and the fracture, again from the photo, I would suggest that your stone is a piece of the sandstone from the mesa caprock in the park. Similar rocks from the area could contain fossils but I don't believe your sample does. In the South Western States sandstone is often cut and displayed for colour patterns similar to what you see in your stone.
A similar pattern can also be found in a rock formed in areas with rhyolitic lavas. It is like a poorly formed jasper and often found with concentric patterns in the stone, it is locally called wonderstone, local to the Southwest States. Though the geology of the park doesn't support finding wonderstone within its boundaries.