The RFEF said in a statement that Pique was also banned for two months from all official football activity.
Following Andorra's 1-0 home loss to Albacete on Friday, referee Alonso de Ena Wolf noted in his report that Pique shouted at match officials in a threatening manner, Spanish media reported.
Other club officials were also involved in the incident, including Andorra president Ferran Vilaseca and sporting director Jaume Nogues. Vilaseca has been banned for four months. Nogues, like Pique, received a six-match suspension and a two-month ban.
The RFEF disciplinary committee also ordered Andorra to close its presidential box and VIP areas for two matches, and fined the club 1,500 euros ($1,762).
Friday's defeat dealt a heavy blow to Andorra's chances of qualifying for the promotion playoffs, with the club sitting eight points below the top six with four games left.