Well, IMHO, on the one hand since it's a doctor that's making the recommendation to keep running - that carries a lot of weight. That being said, there are all sorts of doctors making all sorts of mistakes. So - I personally would discount the recommendation if either 1) the doc is not a podiatrist or 2) the doc doesn't know running very well.
Overall I'm surprised with a recommendation to keep running. A stress reaction means your body can't handle what you're giving it. So continuing to do it while also trying to have it heal doesn't make much sense to me.
Jim