Hello, I'm Dr. Deb and will do my best to help you today.
I'm sorry for this concern about Max.
If he's eating and drinking and otherwise acting like he normally does, then he may have bruised his body when he fell off of the bed. It would take a great deal of force to break a rib so while I suppose this might be possible, it's not highly likely. But, for the sake of argument, even if he had broken a rib, we don't typically try and repair them or wrap the chest like you would in a human. We treat these patients with anti-inflammatory and pain medication and allow the body to heal on its own.
I do have a few suggestions which may be of help, though:
1. Cool compresses can be applied to the side of his body if it's been less than 72 hours since he fell off the bed; if it's been longer than this, then warm compresses are best....but be careful to only apply them for 5-10 minutes at a time. They should be wrapped in a towel or something similar since it's possible to burn the skin if ice or hot compresses are aredirectly applied to a dog's body.
2. As long as he's not vomiting nor taking steroids or nsaid medication, you could give him Aspirin 10 mgs per pound of body weight twice a day; this drug should be given with food to avoid stomach upset. I'm sometimes hesitant to advise it's use since it might interfere with what your vet would want to prescribe (if he has to be seen) but one or two doses will not be harmful. The anti-inflammatory as well as anti-pain properties could help him feel better.
3. Leash walk him only when he's outside for a minimum of 7-10 days if you're not already doing so. Don't let him run around or it may take longer for his body to heal. And, obviously, I would try to avoid picking him up by his chest or ribcage.
I would expect him to start healing in 3-4 days but if he's not and/or he's still sensitive in a week, then a vet visit may be wise.
I hope this helps. Deb