No. Never. Under no circumstances.
We cannot provide your full Credit Card number to anyone, since we do not store it. At checkout you sent it directly to the payment processor of your choice (Stripe, Paypal, Amazon Payments, Compro Pago, etc.). The third party sends us the confirmation that the payment went through, which then allows your order to be processed. Magic!
The last 4 digits of the card are often used to confirm which card was used to place the order. This is in compliance with industry standards and does not allow a bad guy to guess or predict your full Credit Card number. When you chose to receive order updates on Facebook (default is email), your last 4 digits will be sent via Facebook. If you do not like the idea of having this information on Facebook, make sure you do not opt-in for Facebook updates at checkout.