Even if he would say "That happened because I tried to bring her back", it would still be safemto asume that the Stones went crazy because of it. At some point after putting the Gauntlet on ans before he snapped his fingers, Hulk's mind already took the decision of using the Gauntlet (snapping his fingers or in another way) to bring Nat back, idea which the Stones, especially the Soul Stone could not do (I may be wrong refering to the Stones as actual entities, but it seems that they actually are). As a result, the Stones "activated" some sort of backup plan, aka started to burn Hulk's hand till he gave up the idea of bringing Nat back and doing something possible, like bringing the half of the universe back.
Also, if I remember correctly, Tony experiences a power surge too after the Stones get arranged in the Gauntlet patern. The respective power surge didn't burnt Thanos when he experienced it. You could argue that the Power Stone's surge almost killed the Guardians and the Space Stone's teleported Red Skull on the other side of the galaxy, but we don't see Thanos experiencing any of these two (my guess is that the other four stone negate such severe effects, seeing that we see people having the Mind, Time, Soul and Reality stones in their hands and not really getting hurt).
Also, the souls reacted like this for one person instead of half the universe because Nat was the soul sacrificed for getting the Soul Stone. As Hawkeye said (and I think Red Skull told him): there is no way tk bring her back, just like Thanos (or anybody post IW-pre Endgame) could not bring back Gamora using the Stones. On the other side, you can very easily destroy half of the universe, or bring it back, because the Stones are not bounded by that souls. They do not depend on those people. It's like assigning an assassin to kill half of the population of Earth, which means a lot of people that he never knew and doesn't have anything tk do with them, versus killing his own mother. Only that the mother would be much, _much_ important.