Matthew,
I think probably not, certainly for the big chains.
I normally rent thru' Hertz, and their t&c's state clearly that, while you can settle your final bill post-rental with a debit card, you need to have a credit card pre-approved to drive the car away.
It may be that you could find an independent operator who would accept a debit card, but not sure how widespread they (debit cards or suitable operators) are in the US.
Stewart