I leverage this as a "vote with your wallet" factor; small local business I want to support? I'll pay with debit and lower their transaction fees. Chain store I didn't really want to shop at, but they already displaced the local supplier? Credit, and I get 1% back.
If you had a choice between "universe with `reward' programs" or "universe without," then you'd be paying for the rewards. When we're stuck in a universe that has them, and that has unified prices, well then no, everybody even cash customers are paying for the average transaction fee, and the person with the rewards card is paying less.
At employee-owned businesses like regional chains Winco and Bi-Mart, I always use debit and save them the fees.
At locally-owned gas stations with lower prices they'll have a cash price and a credit price. The big chains don't bother, because they have enough volume to pay much lower transaction fees.