The phrase "to support the war" means that someone is in favor of a military conflict and is offering help or encouragement for it to continue or succeed. Full definition
I've been deployed for nearly 5 years supporting our war effort. (religion.blogs.cnn.com)
But there's more: the participants who supported the war were more convinced that there were weapons of mass destruction after they saw the article correcting the mock article. (climaterealityproject.org)
He can certainly support a war politically but the act of declaring a war is outside his «powers». (politics.stackexchange.com)