Yeah, you can but if you leave it on for more than a few minutes it makes the meat mushy.
The active ingredient is papain, the same thing in Adolph's meat tenderizer, commonly available in supermarkets.
Pretty good explanation of the chemistry at Wiki.http://en.wikipedia.org/wiki/Papain
When we lived in South America, I noticed that my cook and housekeeper always kept the papaya rinds to use on their skin. When I asked about it, they told me it worked wonders on wrinkles and keeping skin smooth.
Dang, if it doesn't work as well as anything from an expensive department store counter.
I've got the remains of big papaya in my fridge right now that isn't going into the trash even though it's over the hill for eating. I'll put little tubs into the freezer to use over the next few weeks.
This is not something I would go out of my way to do. As Edward Adams said above, it can give meat an unappealing texture and flavor.
Most times when you see papaya used, it's a relish/salsa of some kind. I'm not a major fan of cooked papaya but that may be personal.
Perhaps we'll hear some other opinions.
You could get some inexpensive cut of pork or beef and experiment. It's only one meal after all. Satisfy your intellectual curiosity.
I'd rather eat the papaya plain with a little lime. That was always the best home remedy for a funny tummy by the way. Fixed it right up. I had those frequently in Latin America b/c I could never resist street vendors.
Papaya is a magic fruit.
Papaya does contain an protein digesting enzyme, one that is used in most commercial meat tenderizers. It is dangerous stuff. It can give beef an unappetizing mealy surface and "livery" taste. My personal preference is to select the right cooking method for the cut of meat that you have purchased. More cooking to get the tougher cuts tender.
That said, I have to admit that I've never understood marinades. Unless it is very thin cut meat, how long do you have to marinade to get penetration?