Astana, which includes Tour de France winner Vincenzo Nibali, will keep its World Tour licence - although on probation - despite five senior and development squad riders being caught doping with EPO and steroids since August.
"It can't carry on, it won't, and it isn't going to," Cookson insisted to the Associated Press in a telephone interview on Thursday.
Still, many cycling commentators and fans called Wednesday's decision a step back, two years after the Lance Armstrong case.
Cookson said the UCI licensing panel imposed strict probation terms, including an audit of the Astana team by a Swiss university which will report in February before the European season starts.
The panel also awaits an Italian prosecutor's file on Astana riders' alleged links to Armstrong's banned doctor, Michele Ferrari.
"This is not the end of the story here," said Cookson, who was not involved in the licence panel's ruling. "The team have been given a very short lead and we will be looking at this matter again. New information is coming to light all the time."
Cookson, who was elected in September 2013 with a mandate to reform cycling after the Armstrong era, said he did not know what kind of evidence could come from the Italian Olympic Committee.
"I only know what I have read on the internet and in the media. We need to see that file," he said, acknowledging frustration at the time the case has taken since the UCI ordered Astana's licence review in October.
Astana's triumphant messages posted on its website and Twitter account, ignoring the probation conditions, also left a bad impression.
"I was personally very disappointed to see the team's reaction," Cookson said. "There is certainly nothing to be proud about, and nothing to be pleased about."
Cookson declined to give an opinion on the target of many fans' anger - Astana general manager Alexandre Vinokourov, a former Ferrari client who served a two-year doping ban before winning the 2012 London Olympics road race.
"It's not my place to make a comment about an individual," the UCI leader said, adding that leaders of the Kazakhstan government-backed team and cycling federation "all need to take their responsibility here."
Another positive doping test by a senior rider in 2015 will also trigger a review of the licence which guarantees Astana entry to top stage races and one-day classics.
"They won't get a second chance," Cookson said. "They have had too many second chances in the past, perhaps."