She's responsible for the war because she told her Captain that killing the Klingon would make him a martyr which would lead to war, and that they should capture him.
But her emotions got the better of her when Captain Georgio was stabbed and she switched her Phaser from stun to kill and...made him a martyr.
Thats not true at all. They were at war already with the Klingons and making him a martyr would just galvanise the Klingons efforts. Michael was also the one that made that claim. The captain wanted to take out the ship completely and Michael was the one who said not to because it would potentially allow them to defuse the situation if they captured him.
The only thing that would have avoided a war at that point was to follow Michaels advice and take out the klingon ship before all the houses showed up. The captain stopped that and war became inevitable at that point. The only reason Michael is blamed for the war is because the writers decided so. There is no logic to it. In reality she would probably be court-marshalled but potentially let off in light of her actions being in the best interests of the federation and the fact that if she was allowed to carry them out there would almost certainly be no war and thousands of lives would have been saved.
As the captain of Discovery said at the end of the episode, what she did was the correct course of action and the unwillingness of the federation to allow captains/officers to think and act on their own knowledge/views was the issue.