The pitch invasion by Hiberian fans at the conclusion of the match their side won 3-2 was followed by Rangers supporters going on to the pitch and fighting broke out.
The Ibrox club says their players and officials were attacked by Hibs supporters.
On Tuesday, the SFA announced that both clubs had been issued with notices of complaint.
The clubs are alleged to have breached disciplinary rule 311, which states “damage was sustained to Hampden... as a consequence of misbehaviour by supporters”.
They have until 6 September to respond and hearing dates have been set for Hibs on 4 October and Rangers a day later.
However, Rangers are furious with the charges levelled at their club and questioned the body’s competence to govern the game.
A statement issued by the club said: “Rangers’ directors are shocked at the charges issued by the Scottish FA’s compliance officer in the wake of the pitch invasion at the end of the Scottish Cup final last May.
“It is alarming that the governance of Scottish football is so lacking that it is impossible for the Scottish FA to guarantee the safety of footballers at the country’s showpiece event. The SFA is either unwilling or is powerless when it comes to taking the appropriate punitive action against the offenders.
“A number of Rangers players were assaulted by Hibernian supporters in broad daylight on the Hampden surface and a repeat of this must be avoided at all costs. That should have been the priority of the SFA. Yet, the governing body insists it is not within its remit to issue any charges let alone punishments for such violent and dangerous misconduct. Instead of player and supporter safety they have placed all their emphasis and importance on the monetary value of a set of goalposts, pieces of turf, and advertising boards.
“Rangers Directors will take time to consider an appropriate and more comprehensive response to redress our grievances but for the moment the board fully endorses the incredulity and anger of all Rangers supporters.”