Yes, you can be fired for not coming to work (transpo issues aside); should qualify for unemployment. If fired for this issue, naturally you'd want to be clear up front when applying for UC that you were let go for being unable to get to work due to lack of transpo options, public or private. Don't leave it to applying online or phone filing if no room to explain, since you can deduce employer may contest state awarding you benefits.
"Due to hospitalization, I am on a written warning."Unless FMLA applies (you and employer being subject/eligible and circumstances fitting -- mere "hospitalization" doesn't cover it because details matter), this wouldn't be relevant.