Celtic have apologised to Hearts after a pitch invasion following Saturday's title decider that the Edinburgh club called "unacceptable" and "disgraceful".
Moments after Callum Osmand's goal , a number of home fans ran on to the pitch to celebrate, with several confronting the beaten opposition.
Hearts said their players and staff reported "serious physical and verbal abuse" during scenes that had "embarrassed Scottish football".
Jambos players were ushered down the tunnel by club officials and pictured leaving the ground still wearing their match kits, with the Tynecastle Park club saying in a strongly worded statement that "the safety of our staff was our prime focus".
Celtic released their own statement on Sunday evening where they thanked their "magnificent support" but apologised to their opponents for the actions of "a number of individuals".
"Everyone at the club congratulates Martin, our players and all our staff on delivering the club's 56th Scottish league title and a thrilling climax to the season backed by a magnificent support," the statement read. "We thank Celtic fans for everything they have done to help the team over the line.
"The club regrets that our victory over Hearts was followed by a number of individuals entering the field of play.
"We again emphasise that there is no justification for this behaviour which, for the vast majority of Celtic supporters, only detracts from the joy of such occasions.
"We also apologise to Hearts for the situation encountered by their players and staff at the conclusion of the game, and for the fact that these events prevented them from saluting their own supporters at the end of an enthralling campaign, to which they have contributed so much."
Celtic said they would "co-operate fully with any investigation".
Police Scotland said on Sunday that they were reviewing CCTV footage and continued to engage with Hearts "to establish if any of their players were assaulted on the pitch."
The force confirmed that four people had been arrested at Celtic Park, with a further 10 arrests in Glasgow city centre and four in Edinburgh. Two police officers were injured - one seriously - and three members of the public were injured.
Police on Saturday said they had dealt with "unacceptable violence and hostility" as fans gathered in Glasgow to celebrate Celtic's title win.
A statement added: "Around 3000 people made their way to the Trongate area of the city following the match, where we faced significant levels of disorder in which officers were injured. Missiles, including glass bottles, were thrown at them when assisting with a medical emergency.
"Specially trained public order officers were deployed and action was taken to restore order, disperse the crowd and make arrests. Extensive enquiries are under way to identify others involved and further arrests will be made."
Assistant chief constable Mark Sutherland added: "The levels of criminal behaviour witnessed, in particular the hostility directed towards our officers, is simply unacceptable and cannot be tolerated. Those involved should be in no doubt that they will be identified and arrested.
"Urgent discussions with Celtic, wider Scottish football authorities and other stakeholders, including Glasgow City Council, is required to achieve a solution that improves public safety, reduces disruption and prevents this from happening again."
Heart of Midlothian utterly condemns the shameful scenes at Celtic Park this afternoon which have, once again, embarrassed Scottish football.
Reports of serious physical and verbal abuse towards our players and staff, both on the pitch and elsewhere, are deeply disturbing. We are investigating this fully and are in dialogue with Police Scotland.
We will make no further comment at this time other than to say that it is completely unacceptable that our players and staff were put in that situation.
Given the menacing and threatening atmosphere inside the stadium, our entire staff had no alternative but to leave immediately, without undertaking post-match media duties. To our media partners, we apologise but the safety of our staff was our prime focus during these unacceptable scenes.
The pitch invasion caused a chaotic end and nobody seemed to know whether or not the match had been brought to a finish. Our players were then denied the opportunity to thank our magnificent fans - sensational to a person - for their backing this afternoon and all season long.
We expect the strongest action possible to be taken by the footballing authorities in the interests of protecting the safety of players and supporters, and the integrity of our game.
This season has seen Hearts capture the imagination of football fans, not only in Scotland, but the world over. We pay tribute to Derek McInnes, his staff, the first team and the supporters, who have done the club proud this season. None of them deserved the disgraceful scenes that unfolded.
Former Scotland striker James McFadden said in the immediate aftermath on Your Site: "This is unacceptable. The players should be safe on the football pitch. You do not want to see it. There will always be people spoiling it for everyone else.
"You can hear the reaction from the majority of the Celtic fans in here with the fans on the pitch, the booing.
"You don't want to see it. It's a game of football, you have got two teams that have earned the right to go toe-to-toe on a day like this, and the winner emerges with the trophy. Hearts have played their part by a huge stretch, and you don't want to see overspill of fans.
"I understand the emotion of the occasion; you have a right to celebrate but you can't cross the line. If that is right that Hearts players have been assaulted, that is absolutely unacceptable."
Your Site' Chris Sutton added: "There's no place for it. You can imagine how the Hearts players felt after such a long season and then losing at the end."
When asked about the scenes after the game, Celtic boss Martin O'Neill said: "If some Hearts players have been accosted, then that's just not right at all.
"I genuinely didn't see the scenes at the end because I thought the game wasn't over, so I'm trying to push some of our fans off the pitch again to play whatever's left.
"Obviously, at 3-1, I think that the fourth official had said to me, there was like a minute to go when we scored the goal, so with the best will in the world, they're not going to get two goals in that time.
"Obviously, there's a lot of commotion in the tunnel, so I don't know, but getting back to the point, if Hearts players have been accosted by some of our fans, that's pretty serious. Honestly, I'm not pushing it to the side, it'll be something that the officials or people will have to look at."
