Yes, in the sense that they can't charge customers paying with a credit card a different price. However, because payment fees are an expense to their business, merchants absolutely pass on that cost to customers - it is reflected in the final price of whatever they are selling. In a sense, the non-credit paying customers subsidize those paying with a credit card.
Of course they/we do. But, the Nash equilibrium is at 100% credit card use: if I stop using credit cards at vendors, I lose out on the benefits (interest-free loan, chargeback ability, rewards, purchase tracking through Mint), in return for a savings of 1 - N/(N-1) * transaction fee = zero. I have an incentive to keep using charge cards.
But merchants are allowed to give customers a discount for paying "cash" (actual cash or debit as the debit fee can often be passed onto the customer).