The big point of ghee is that there are NO milk solids in it when prepared correctly- that means all that is left is fat! With no sugars or proteins left for bacteria to feed on, it is as safe (theoretically) as leaving olive oil or coconut oil at room temperature.
The jar that has been in the fridge- take it out of the fridge if it's relatively new, otherwise prepare to never use it. If it's inconvenient, in my experience, it just sits there while I spoon out the coconut oil. If you do have a "wet" fridge (does water condense on every surface? Is there a lot of moisture on the shelves? Is your fridge quite old?), mold could be a concern, especially if you've been using contaminated utensils (read- dunking knives that haven't been freshly washed in). In that case, when you do want to use it, you could take out a small amount at a time to bring to room temp the old fashioned way, or if you have a microwave yay for short cuts. If mold does develop, because it has such a high fat content, you can "treat it like cheddar"- excavate out the moldy section, leaving a few good inches on each side to ensure that you get all the roots. Even in my food preparations university class taught by a microbiologist, this is considered pretty darn safe. If you don't feel comfortable doing that, you could toss the lot and keep it moisture free next time!
For new jars: no need to refrigerate. Probably safer not to, just because of the typically higher moisture content of most fridges compared to pantries.