I think what was said (and how it was said) matters here. I tend to fall asleep easily when I'm not driving a car too. But I also know that when someone is driving early in the morning/late at night it's nice to have someone to talk to vs. someone asleep beside you. You do have to get up early in order to meet your coworker for the carpool. So does she. And on top of that, she has to stay alert and drive for an hour. I can understand her getting resentful that you sleep for that hour while she drives.
That being said, she needs to make an effort. I'm not sure why the arrangement is so that she drives 100% of the time (do you have a car?), or how that came about. You seem to be meeting your end of the deal as far as paying for half the gas, etc. If she wants you stay awake and talk to her, then she should tell you this politely. Not argue. Asking you to help keep her alert, IMO, is not an unreasonable request if it's requested politely.
It kind of sounds like she has been resentful of this for a long time and let it build up to the point that it caused an argument. Ideally, you'd be able to grab a cup of hot coffee early in the drive, and have an engaging conversation...keeping both of you awake. But, if this isn't possible, then I agree that you need to find another way to get to work. This arrangement isn't working for either of you.