The Wrangler tailgate hinges are marginal at best. An oversized spare will inevitably lead to problems. The hinges sag, causing misalignment of the door. The tailgate itself can bend from the weight. And bouncing on the trails will make it much worse.
There are several solutions. Which one is best is (and likely always will be) the subject of heated debate.
The first option is heavy duty hinges. This can keep your hinges from getting saggy, but I think 35's are about the limit. And they do not prevent the extra weight from altering the shape of your tailgate.
THIS is an example.
You can add reinforcements to the tailgate, like
THIS, to prevent the sheet metal from bending, but it's still limited by tire size.
The next option is a pivot-type heavy duty tire carrier from someplace like Smittybilt, Terraflex, et all.
THIS is what I have used on both our current Jeeps. This transfers the weight of the spare to the body tub, which is considerably stronger than the sheet metal of the tailgate. It also allows your tailgate to continue to operate normally; there are no extra steps involved when you open it. The one I linked is good for a 37" spare. You could go larger, but your bumper will be in the way.
The final option is a swing-away tire carrier, which you can find in body-mounted or bumper-mounted versions.
THIS version is body mounted. It's limited to 35" tires, same as the pivot-type carriers. It mounts to the body tub on both sides of the tailgate, which does spread the weight out over a larger area. This would be vital if the single-side carriers were bending the tub, but I have not heard of this being a problem.
THIS version mounts to the bumper. It transfers the weight of the tire to the frame, which is obviously the strongest possible support. This makes it the best option for really big tires. With this option, you need to either buy the matching bumper or be willing and able to modify your current bumper to work with the carrier. Welding will be involved, and obviously an OEM bumper is not going to work.
Both styles of swing-away carriers also require more work to open the tailgate. You will no longer be able to just grab the handle while juggling groceries in your other arm. On any kind of slope, opening the tailgate can be a two-arm job.