Special Counsel Independence and Integrity Act (Sec. 2) This bill amends the federal judicial code to codify certain Department of Justice (DOJ) regulations that govern the appointment, oversight, and removal of a special counsel, including regulations: that authorize the Attorney General to appoint and select a special counsel; that establish the qualifications, scope of jurisdiction, and authority of a special counsel; and that authorize the Attorney General to remove a special counsel for misconduct or other good cause. Additionally, the bill provides new statutory limitations and requirements with respect to the removal from office of a special counsel. Specifically, the bill: requires removal to be by the Attorney General or, in certain circumstances, the most senior Senate-confirmed DOJ official; requires a special counsel to be provided written notice that specifies the reason for removal; and allows a special counsel to challenge the removal in federal court. The limitations and requirements with respect to removal apply retroactively in the case of a special counsel who is appointed after January 1, 2017, and removed before the date of enactment. Finally, the bill requires DOJ to notify Congress when a special counsel is appointed, before a special counsel is removed, and after a special counsel's investigation concludes.