If both the wheel and axle gears are fine, remove the clip that retain the axle gear and check to see why the pawl isn't engaging or that possibly the gear was installed backwards.

If we are referring to the small drive gear at the end of the axle, where the wheel is not driving- then we are meaning the same thing. The axle shaft is solid from end to end in this transmission. If one gear is able to turn its wheel- fine then that shaft and the transmission itself is functioning.
"Thanks for the recommendations. I'm assuming you're referring to the square/rectangular-like box that's on the underside of the mower where the axle passes through when you mention "remove the CLIP that retains the axle gear", correct?"

Clean and lubricate the ratchet inside the small gear on the end of the axle. There is a hole in the end of the axle for cleaning/lubricating these assemblies and it should be done every year. Take a toothpick/pipe cleaner and see if you can clean out the hole some before shooting anything in there. One of the best ways to clean it without disassembly is to use an aerosol spray solvent. I like 'Gun Scrubber' because it's safer on paint, etc. than brake or carb cleaner, but you can use those too. Spray the gear, turn it and spray more, particularly at the rear where it rubs on the axle. When you think you have it pretty good, now shoot it right into the lube hole in the axle. Keep cleaning until you hear no grittiness when turning the gear and you do hear the ratchet clicking. Now you can use the aerosol can of White Lithium Grease with the straw and shoot that into the lube hole. While wet, it also carries a solvent and will probably show some more dirt coming out. Repeat until clean. You might as well do the other side too before it locks up on you. Once you do this, the next time you probably can just use the spray grease. In fact, the spray grease alone might do the job - it depends on how cruddy the ratchets/springs are in there. Also clean the axle bolt, axle bushing inside wheel hub and driven gear on the wheel. Lube these with #2 NLGI grease or the spray lithium grease. If you have a grease gun (might be a good idea to get one of the small grease guns like at Wal Mart) you can inject grease into the lube hole after cleaning. My favorite mower wheel grease is.....don't laugh......Slick 50 'One Grease'. It's a #2 Teflon grease, green in color and sometimes can be found in little 1 oz tubs as well as the 14 oz gun tubes.
Hope this hasn't been too rambling - I just wrote off the top of my head.
It's a very messy job. I put a piece of 4x4 under the rear of the deck which gets both wheels off the floor and put a piece of cardboard or some newspapers under the rear of the mower. I'll put some solvent in a plastic pan and have a small paint brush, a toothbrush and some rags there for the job. When you take a rear wheel off, be aware of washers and how they are re-installed.
If you can't get the ratchets to work in this way, you'll have to pull the gear as tomplum said, and replace pawls/springs as required. But chances are this 'in-situ' cleaning will fix it.

Ended up being part # 66-6050 (left-hand rocking key) that was bad on my mower. Replaced now by a local small engine repair shop and all seems to be fine again with both rear wheels engaged when self-propel bale is gripped.