It is indeed possible you hurt it at the gym, though I have never hurt my back on benchpress, even while arching my back.
How far must you have arched it to hurt after benching? The reason I ask is, the bench press itself won't put pressure on the back at, it should be directly on the shoulder blades when arching thus only the chest and triceps really work at pushing the weight.
Thus if it is the back muscles that hurt, you must have pulled too hard to arch your back, or you did another back exercise that hurt the back muscles. If it is the spine that hurt...you need to work flexibility somehow, because the spine can actually handle a huge amount of weight even while not straight. Think about heavy deadlifters for example.
NBMy guess is, your back muscles was tired after the gym, and then whilst playing pc games sitting in a bad position, the muscles was too tired to assist the spine and protect it from the bad position, and now the spine is hurting.
Lesson you hopefully learned is, after doing super heavy exercises at the gym, be careful how you treat your body after the fact for that same and next day. Sit up right properly with your back assisted by the chair, because your muscles will be too tired to assist your body. This is important after deadlift and squat days, and somewhat after bench and overhead press days where you core works quite a bit too.
That said, you should feel fine after a day or too, if not go a bit lighter on your back in the gym and treat it nicely until it feels better.