"If my U.S. application claims priority to a European patent application, will I also have to file with the USPTO a paper copy of the priority document claimed?"
No, the requirement for the certified copy of the foreign application will be considered satisfied if the electronic copy of the foreign application is received by the USPTO within the period set forth in 37 CFR 1.55(a) (e.g., before the payment of the issue fee).
However, because successful retrieval cannot be guaranteed, please be sure to monitor PAIR to confirm that the electronic copy of the foreign application has been received by the USPTO. At the time of allowance, the examiner will provide an acknowledgement on the Notice of Allowability (PTOL-37). If the priority document is still missing, applicant must file a paper certified copy of the priority document prior to the issuance of the application. See 37 CFR 1.55(a).