Amsterdam Police told Press Association Sport that fans armed with bottles and sticks attacked plain-clothed police in an incident described as "coming out of nowhere".
Eight police officers were injured with one knocked unconscious following the fighting in Dam Square, in the city centre.
Police said the majority of those arrested were Celtic fans, although it is believed that supporters from other clubs were also involved.
"At the end of the afternoon a large group of Celtic supporters attacked police officers in plain clothes," a spokesman told Press Association Sport.
"Eight were injured and one was knocked unconscious. A few of them had broken noses and needed stitches above their eyebrows and on their lips.
"Bottles and sticks were used in the attack which came out of nowhere. There were 15 arrests, mostly Celtic supporters."
Celtic lost the match 1-0 and face an uphill battle to reach the knockout stages of the competition.
Neil Lennon believes some of his players departed the Amsterdam ArenA "with regrets".
The Scottish champions were far from their best in the crucial Group H encounter and were sunk by a 51st minute strike by Lasse Schone who had scored after coming on as substitute in the Dutch side's 2-1 defeat at Celtic Park.
The defeat consigned the Hoops to the bottom of the table with three points from four games.
Ajax moved above the Parkhead club on four points with Barcelona having qualified on 10 points following their 3-1 win over AC Milan at the Nou Camp. The Italians visit Celtic Park later this month before Celtic play Barcelona away in December.
While there are still various permutations available, Lennon's men will be out of the Champions League if they lose to Milan. The Parkhead boss admitted the match was an "opportunity lost".
He said: "Some players will have regrets because they were nowhere near their best and at this level you can't have that. We could have done better with the final ball and that has been a concern for the last few weeks.
"We missed some decent opportunities, particularly the James Forrest one when were two-on-one. You were thinking 'goal' and that would have got us back in the game.
"Goals change games and from their point of view it was good but from our point of view it was a poor goal. When Schone scored we were just getting a foothold in the game.
"It could all flip round if we beat AC Milan and Barcelona come here and take something off Ajax. But it is imperative that we beat Milan at Celtic Park if we have are to have any chance of qualification."
Ajax coach Frank De Boer believes Celtic still have a chance of making it out of the group with Barca.
He said: "Celtic still have a chance, they are still only two points behind Milan. There was more pressure on us after the game in Scotland so we are relieved."