Vittek, 33, who is Slovakia's all-time top scorer with 23 goals, is injured.
Liverpool's Martin Skrtel is included in the squad after his injury-hit campaign in the Premier League.
Slovakia face Wales, Russia and England in Group B at the finals in France, with their first game against Chris Coleman's side on 11 June.
"The players' names for the nomination were clear so I had no reason to speculate," Kozak said.
Goalkeepers: Matus Kozacik, Jan Mucha, Jan Novota
Defenders: Peter Pekarik, Milan Skriniar, Martin Skrtel, Norbert Gyoember, Jan Durica, Kornel Salata, Tomas Hubocan, Dusan Svento, Lukas Tesak
Midfielders: Viktor Pecovsky, Matus Bero, Robert Mak, Erik Sabo, Juraj Kucka, Patrik Hrosovsky, Jan Gregus, Stanislav Sestak, Marek Hamsik, Ondrej Duda, Miroslav Stoch, Vladimir Weiss
Forwards: Michal Duris, Adam Nemec, Adam Zrelak

They said 11 of the bodies had been buried about two years ago in a desolate area about 40km (25 miles) southeast of Ciudad Juarez.
The remaining eight victims, who had been tortured and killed only days ago, were found along a road near the town of Rosales.
Chihuahua is one of the states hardest hit by drug-related violence in Mexico.
All the victims were male and had been shot dead. Many also showed signs of torture.
According to figures released earlier this year by Mexico's National Human Rights Commission, 16,000 bodies remain unidentified and a total of 24,000 people are missing.
Most of them are believed to be victims of fighting between rival drug gangs or the military response to the drug violence.

University of Aberdeen researchers found the link remained even after taking account of the smoking habits of the women themselves who miscarried.
But no link was found between exposure to cigarette smoke in the womb and a decrease in fertility.
The study examines data from 12,321 women born before 31 December 1972.
The university said an analysis of the data also showed that women exposed to cigarette smoke in the womb were more likely to have a pregnancy than those whose mothers did not smoke.
Women exposed to cigarette smoke in the womb were also likely to have a pregnancy earlier than those not exposed.
Dr Sohinee Bhattacharya, a pregnancy and childbirth expert from the University of Aberdeen, said previous research had suggested there may be a "small decrease" in the fertility of women whose mothers smoked in pregnancy.
"This study used a databank in Aberdeen to look at the birth records of more than 12,000 women and followed them through for 40 years to see whether being born to a smoker had any impact on women's future fertility," she said.
"The study did not find a link between mothers' smoking and a decrease in fertility in their daughters.
"Women whose mothers smoked were more likely to have a pregnancy, but this study suggests that this could simply be linked to the fact that they were also more likely to get pregnant at an earlier age and could be related to socioeconomic status.
"Worryingly the study did find a significant increase in the chances of having a miscarriage among the women whose mothers had smoked when pregnant with them. "
The lead author of the paper, Sam Tweed, completed the work as part of his undergraduate degree and graduated from the University of Aberdeen's Medical School in June.
The research was published in the Human Reproduction Open journal.
Striker Robert Vittek is the headline absentee from the provisional 27-man squad Slovakia coach Jan Kozak has named for Euro 2016.
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Mexican security forces have found 19 bodies in mass graves in the northern state of Chihuahua.
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Women exposed to cigarette smoke while in their mothers' wombs are more likely to experience miscarriage as adults, according to a study.