You can't import/export food from one country to another. Which translates as- you can't take food off the ship or bring it back on to the ship. That is against Customs & Border regs. Here's a link http://www.cbp.gov/xp/cgov/travel/vacation/kbyg/prohibited_restricted.xml
The cruise ship has a huge rubbish bin located on the way out and every cruise we take many, many people have to dispose of their would-be lunches. Bottled water should be fine. Because it is a tourist city, you should be able to buy it at any shop in port, assuming Cozumel is similar to other ports.
We have been to Mexico a couple of times. Two weeks in Vallarta with a relative. Salad rinsed in tap water, drank tap water, etc. I have the world's WORST digestion and I had no problems. If in doubt, I might suggest finding nicer place and eat cooked food. There should be a couple of chains from which to choose. Eating at a chain might be somewhat dull, but it might the lesser of two evils...