And if you do then appease them, you'll face an equal but opposite backlash for appeasing them. If America's not headed for a civil war, it's definitely headed for a society where red people patronize red businesses, blue people patronize blue businesses, and everything else is a people boycotting or third-parties not caring to play.
Headed for? It has been that way for decades if not centuries or millenia! (Eastern Roman Empire sports politics say hi.) There aren't exaxtly many self proclaimed communists hanging out at Bubba's BBQ Bar and Grill - with demographics always comes politics implicitly.
Worrying about it is the same undue hysteria as thinking Citizen's United means that Pepsi and Coke support is more significant than any voter issues because they have large ad spends.
That's why I wrote "If" and " influence your bottom line ".
It's practically just turning it upside down: instead of assuming twitter needs to be paid attention to because it's twitter, it is far more practical to say: "things that have a significant influence on us/our bottom line/our brand need to be paid attention to". If there is a case where that's a twitter mob, then that is a case. If not, then you ignore them.
The only way to deal with an outraged Twitter mob is to ignore them. They can never be appeased.
It seems to me like Twitter is the new Tumblr. Full of crazy people.