The 24-year-old was issued with a notice of complaint by the association offering him the suspension and it has been accepted.
Ross will consequently miss this weekend’s home league fixture with Stenhousemuir and next week’s trip to Stranraer.
Peterhead have also confirmed they have disciplined the former Aberdeen youth after reviewing video evidence of the incident.
It read: “It is clear Scott did spit in the general direction of Rangers player Nicky Law and Scott admits that he did spit, but in no way was it intentionally directed towards Nicky Law and [he] is making every effort to speak with Nicky to apologise for the misunderstanding.
“Peterhead FC, regardless of the circumstances, cannot condone spitting in any way and, although we accept Scott’s explanation, he has fallen below the level of behaviour we expect, and has been fined under the club’s disciplinary code of conduct.”