If you'd said to the white player no he couldn't return if he had gone off, what would you have done if he had gone and sat in his in-goal area for a few minutes before rejoining the game? Presumably nothing - but he has had his few minutes rest.
as for yellow I guess you are just left forever now ascertaining exactly what the reason is for any player going off. However,as he was clearly in pain, and there was no blood its difficult to see what else it was except an injury. If he needed only a rest he could also have gone and sat in his ingoal area...
didds