Obito was ready to die for his friends trusting the life of Rin to Kakashi, then he comes back to aid them just in time to see him betray his promise and kill her with his own hands. Kakashi didn't just let her die, he killed her for the mission after all the trouble they went through the Gaiden. It is not just for a girl, it is because his idealistic view of the world is shattered, he witnessed the real world is cruel and doesn't have a place for his Naruto-esque idealistic viewpoint and the only way to fix it is with the Moon's Eye Plan.