Generally the answer is no, you cannot move money from one credit/debit card to another directly. These are not designed to be peer to peer payment methods, they work through centralized settlement networks.
There are ways to push funds into credit/debit cards through these networks. For example that's how you get merchant refunds or charge backs. But these still go through the processing network.
So if you're opening a store and you want your customers to pay with credit/debit cards - you'll need to have a merchant account to process these payments. You will not be able to avoid a payment gateway.
Separately, you may be able to set the merchant account so that all disbursements go to your debit card, if your processor supports that, but that would be independent of the purchase transactions made on your store by your customers, and they wouldn't be aware of that or care.