For construction of a second story screened porch with a gable roof, is it acceptable to attach the floor joists to the ledger board by resting them onto a 2x2 cleat nailed into the ledger and also toe-nailing the joists into the ledger? Can this system safely carry the load?

Please don't do that.Use the right sized joist hangers for your joist.
How are you attaching your ledger board?

I am sure the first response to use joist hangers is a good idea and I think they would be worth the small price for piece of mind, but I do "think" that a 2x2 ledger is allowed. I wouldn’t use one either though. I recently framed a floor. I used 2x12s for the center girder and 2x10s for my floor joists. I used a 2x4 ledger and therefore only notched 1 1/2 inches out of each of my joists. I also measured exactly how much to notch out of each 2x10 because I wanted to be sure that each floor joist was resting firmly on the ledger (You know lumber often varies in size up to 3/8s of an inch!). Joist hangers may be best but I felt like this was pretty sturdy. Good luck

Dave