He fought an Iwa-nin after that while using Sharingan and ended with a Chidori if I recall correctly. Medical nin could save him like ... RINIt was crushed badly, they were in a dire situation, and Kakashi was probably low on Chakra. They probably figured they needed their chakra and he was going to die no matter what.
And yes, he did live. However, his entire right side is gone, and was replaced with Hashirama's cells. So it's safe to say no one could have saved him besides Madara. Plot or not, if Kakashi had used Chidori then Obito would have died anyway.