Celtic chief executive Peter Lawwell on Thursday condemned the "vile, sustained and relentless attacks" on his club on Thursday.
On the same day a man appeared in court charged with assaulting manager Neil Lennon during a match, a bullet was found in mail addressed to the coach.
Lennon was attacked by a fan during Wednesday's win at Hearts. John Wilson, of Edinburgh, was charged under Scotland's anti-sectarian laws with assault and breach of the peace, both aggravated by religious prejudice. He made no plea during a closed court appearance.
A request for bail was rejected and he was kept in custody.
Sectarianism continues to plague the Scottish game. While Celtic is traditionally supported by Irish Catholics, Glasgow rival Rangers and Hearts, based in Edinburgh, are both mainly supported by Protestants.
Police confirmed Thursday that a bullet was found in mail addressed to the manager at the club's Celtic Park ground. It was removed for forensic tests.
In a strongly worded statement posted on Celtic's website, Lawwell said: "It is intolerable that any football club, or individual, going about their lawful business in the name of sport should be subjected to this ongoing campaign of hatred and intimidation.
"This is Scotland's shame and it is high time Scotland addressed it."
Police also arrested two men on Thursday in connection with the mail bombs sent to Lennon and two prominent supporters of the Glasgow club in March and April following an ill-tempered match between Celtic and Rangers.
The devices were intercepted before reaching their targets and never exploded. The men, ages 41 and 43, were detained following raids in Kilwinning, 30 miles southwest of Glasgow.
Earlier this week, seven people were charged with possessing a fake firearm near Celtic's training facility.
"Since moving here a decade ago, Neil Lennon has had to endure prejudice and violence both as a player and manager, having suffered no such problems elsewhere," Lawwell said. "We are deeply appreciative of the intensive police efforts to address the criminal offenses that have come to light. All right-minded people will surely condemn these actions but, as a society, we must also address the underlying factors that lead to such behavior."
Scotland's First Minister Alex Salmond insisted the country is moving toward banishing sectarianism from the Scottish game.
"Throughout society substantial progress has been made," Salmond said, citing the successful visit by Pope Benedict XVI last year. "But quite clearly surrounding and attaching themselves to the game — let's not talk about them being part of football — we have some bigots and fools attaching themselves to the game."
Lennon has been targeted throughout his career. He quit playing international soccer for Northern Ireland in 2002 after 39 appearances, saying he had received death threats from a paramilitary group in his home country.
Earlier this year, Lennon and players Paddy McCourt and Niall McGinn, who are also from Northern Ireland, were sent packages containing bullets.
Wednesday's incident came in a televised match, which Celtic won 3-0 to move within a point of league leader Rangers heading into this weekend's final games.
After Celtic scored the second goal, Wilson ran onto the field from the home section of the main stand at Hearts' Tynecastle stadium. He appeared to lose his balance as he lunged at Lennon, but still made contact with his left hand to the back of the manager's head.
Wilson was wrestled to the ground and taken away by police.
Lennon did not speak to reporters after the game, but posted on Twitter: "Don't let what happened to me tonight take the shine off a wonderful team performance... I don't walk alone."
His assistant, Johan Mjallby, said Lennon was "shocked."
"We have to look into this because when you work in management you should be secure in a football ground," Mjallby said.
Hearts announced on Thursday that the fan has been banned from the club for life. The club has launched an investigation into the incident.
The Scottish Football Association is also looking into the attack, particularly in light of the breach of security.
"Clearly this kind of behavior from supporters is wholly unacceptable," chief executive Stewart Regan said. "The safety of players, club officials and match officials is paramount on or around the field of play.
Lawwell said he was convinced the incident had wider ramifications than the soccer pitch.
"Last night's appalling attack on Neil Lennon brings shame on Scottish football," he said. "And it again highlights the fact that Scottish society must address fundamental and serious issues which lead to outrages of this kind."