Can I sue my disability company for causing me to get fired from job?

I work in State Farm’s call center and New York Life is the third party that handles the short term disability for State Farm. I was in a house fire while at a friend’s house and due to the smoke and running back in to pull people out of the fire before the fire department arrived, I received damage to my vocal cords and some damage to my lungs. I am still having issues with my vocal cords and I am trying to heal. I have been following my general practitioner’s orders and staying on medical leave until after I see the ear, nose throat doctor again. I have been providing all the information requested by New York Life as soon as I find out, and have been doing everything by the book. However, New York Life closed the claim within 48hrs of receiving the contact information on the ear, nose, throat doctor, and then never requested anything from said doctor like they said they would before passing judgement. My job is threatening to fire me due to my absences not being protected since New York Life has denied the claim. If I am fired, is it possible for me to sue New York Life for costing me my job of six years and my health insurance needed for me to heal?