Rear wheels- When the gas pedal is pushed on the John Deere Gator, should both wheel engage and spin? Only one rear wheel turns when gas pedal is pushed. No sure if this is a problem.

https://partscatalog.deere.com/jdrc/sidebyside/equipment/83784/referrer/navigation/pgId/17379192
Above link is the MFWD parts diagram for your picture (orientation is same as well). Yours looks ok, from what I can tell looking at the picture.

With MFWD disengaged, if you raise either front wheel off the ground and then spin the wheel, this shaft should rotate and likewise rotating the shaft will turn the wheel. I expect you will find this true, which means problem is with engagement in the rear. See following link.
https://partscatalog.deere.com/jdrc/sidebyside/equipment/83784/referrer/navigation/pgId/17441008

When properly engaged in rear, you wont be able to turn the shaft. Follow selection lever and linkage and be sure its making full range of motion.

Check under the cover of the self propel unit to see if the belt is off the bearing or broken.

are you sure both wheels are spinning in all of the gears? I'm guessing you have a bad motor, you can order a new one here http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=150894216724&ssPageName=STRK:MESE:IT

are both of the rear wheels spinning when you push the foot pedal, lift the rear end when you verify this

if you lift up the rear end and hit the foot pedal do both wheels spin?

do both wheels spin in all of the gears, if not you probably got a bad motor

you can get them on ebay but be careful cause that will put more pressure on the gearboxs

remove the center cap then bend the metal push nut with a needle nose pliers and pull it off then pull the wheels off leaving the axle in place

first of all make sure that both wheels are turning in all gears, you might have a bad motor

1. Remove the center caps from the rear wheels
2. use a 13mm wrench on each side and loosen the bolts
3. remove the rear wheels