Most dogs can safely eat both brown and white rice without serious repercussions. The reason lots of owners prefer brown rice in dog food is that white rice goes through more processing. As the germ, bran, and husk of whole brown rice are removed, a lot of fiber, vitamins, and minerals get lost, too. The milling process also reduces the amount of breakdown that must occur before your Goldendoodle can absorb sugar from it, so it will raise his blood sugar much faster.
Essentially, white rice is not ‘harmful', but it has fewer nutritional benefits for your pet. This, and the fact that it is often cheaper than brown mean it is sometimes considered a ‘filler' ingredient in dog foods.