Your doc says it's OK because a rib injury is one of the easiest to run with, for many reasons:

(1) You have nothing to gain by sitting around. Whatever tiny amount of pain you save now, you will pay for in lost fitness later.

(2) Any other exercise or cross training is likely to involve the ribs as much or more, and be just as painful.

(3) Keeping yourself moving will help to increase healing circulation and "model" the repair of your ribs and surrounding tissue, so you are less likely to experience secondary issues later.

(4) You are going to sneeze and cough anyway, no matter what you do.

(5) It will help increase your pain threshold, one of the most important mental roadblocks on your way to being a successful competitive runner.

There are more reasons, even some for not running, but they are far outweighed by these advantages. One caution I would emphasize is to avoid any kind of running that increases your risk of aggravating your injury with bad footing or another slip and fall, at least until you have a month or so of healing under your belt.

Good luck!