I can't related you injury to climbing but back when I was actively practicing jiu jitsu I torn a section of bone on my pinkie toe that a ligament was attached to. It was a full tear of the bone, took a long time to heal, and that section of the bone became huge during the mending. It stayed that way for a long time afterwards and is still larger than the other. But when I injured it I could pull the toe completely sideways at the joint. If you can't do that, even if it is the bone itself, I can't see it being completely torn. Which is obviously a good thing.
I'm sure someone else with have experience with injuries like this that directly relate to climbing. My only current climbing experience is some soreness in my finger joints (particularly when force is applied sideways to the direction of the joint), but nothing as bad as you describe. I'm taking a few days off because of it though.
Diagnosis either via self or internet is impossible but I would say that you don't have enough information to know that it is a stress fracture or not. Other things can cause pain in that joint and it is common source of pain in climbing. I would just tape it to restrict movement and don't climb or play video games for a week or two then see how it feels. You can ice it too if you want.
Also, not to be an asshole but I would avoid climbing in the gym too much in general. They are notorious sources of injuries and even experienced gym climbers often avoid going more than 2-3 a week. many climbers have sworn them off entirely. Probably the only thing worse on your fingers and forearms than too much time in the gym is too much time on a computer. The combination is, well. . .