I think Michael did it to both Carlo and Tessio so he could be 100% sure.Yes Vito did warn him earlier about the person who approaches him but maybe Michael just needed to know first.Carlo on the other hand,i think Michael just wanted him to actually admit to it before killing him.By both Tessio and Carlo admitting their betrayal it gave Michael more satisfaction.

The hero and the coward both feel the same thing, but the hero uses his fear, projects it onto his opponent, while the coward runs. It's the same thing, fear, but it's what you do with it that matters. Cus D'Amato