I left pickled ginger left out of fridge
Food safety guidelines would say a definite no. The temperature "danger zone" is between 40 degrees F (slightly warmer than your fridge temp) and 140 degrees F. Assuming you keep your house around a normal temp your ginger was most definitely in that range for at least 2 hours (the max allowed time a food is considered "safe"). There's no way to guarantee that the pickling liquid was acidic enough to keep it safe, especially since you don't know if the original label said that refrigeration was required. Commercial products that are pickled and canned are not the same as an opened product, because the canning process creates a vacuum (the lid "pops" when you open it).

All that being said, you may keep it and eat it and never get sick. That's up to you to decide, but for peace of mind, I'd throw away what you have and spend the extra few dollars to buy a new jar.