No, you cannot accept debit cards as debit without a debit terminal connected to the register terminal.
You can accept a debit card as credit via the MSR on your register terminal, but if you accept debit, the PIN number must be entered on a pinpad device as it uses a special encryption method to protect the customer.
Same rule applies to accepting debit if EDC if offline. You cannot accept debit offline, only credit transactions. Debit cards accepted without a PIN number is accepted as credit.