Bank of America Corp. has dropped its plan to charge customers $5 a month for making purchases with their debit cards, responding to a customer backlash after the fee was announced in September.

The move is the highest-profile retreat on the issue, following decisions last week by Wells Fargo & Co. and J.P. Morgan Chase & Co. to drop customer tests of the new fees. SunTrust Banks Inc. and Regions Financial Corp. also said Monday that they will stop charging customers for debit-card transactions.

Bank...