The Scottish FA says referee John Beaton and his family “spent last night at home under police surveillance following a leak of personal details online”.
The association says that it is calling for “tolerance and perspective to prevent any further, unthinkable escalation”.
And Police Scotland have confirmed a man, 19, has been arrested “in connection with a data protection offence”.
Beaton has faced criticism following the award of a late penalty, which was converted, in Celtic’s 3-2 win at Motherwell on Wednesday.
The incident at Fir Park, which involved VAR, was the latest in a series of controversial refereeing decisions as the Scottish Premiership title race reaches its conclusion.
Celtic’s win kept them within a point of leaders Hearts. Those two sides meet at Celtic Park in Saturday’s final top-six fixtures.
“The Scottish FA condemns in the strongest possible terms attempts to compromise the safety of match officials,” said the SFA, which organises refereeing in the SPFL.
“Such vigilantism, motivated by decisions perceived to be right or wrong on a field of play, is a scourge on our national game and we are grateful to Police Scotland for their swift intervention.
“As we approach what should be an exciting finale to the season, we ask those who have personalised and hyperbolised their opinions, those who have sought the easy way out by attributing defeats to perceived refereeing errors, and those who have approved incendiary statements and posts to reflect on their contribution to creating an environment of intimidation, fear and alarm.”
Police Scotland said in their statement: “Officers investigating a complaint of personal information being shared online relating to a Scottish football official, have arrested a 19-year-old man in connection with a data protection offence. Enquiries are continuing.”