Slovenia's three penalties saved by goalkeeper Diogo Costa spelt the end of their tournament after an excellent defensive effort saw them hold Portugal to a 0-0 draw after extra time.
"Of course, they're sad today, and this confirms my viewpoint," Kek said.
"It would be pathetic to speak about how they are heroes and so on and so forth.
"No, those are players who have created a national team culture that we have been missing in Slovenia.
"They have proven themselves - even those who are not playing the games - what it means to be playing for the Slovenian national team.
"I am the smallest character in this team. You have no idea what a privilege it is to be leading such a national team and that we can play the biggest teams in Europe and the world."
Striker Benjamin Sesko had two one-on-one scoring chances in the first 90 minutes, the first he scuffed wide and the second saved by Costa will surely long torment the 21-year-old.
Slovenia, who are 51 places below Portugal in FIFA's global rankings, sputtered in the shootout with Costa saving spot kicks from Josip Ilicic, Jure Balkovec and Benjamin Verbic.
Kek was indignant when asked if the team had practised penalties.
"Don't ask such stupid things," he said. "One can do 30 penalty kicks and then one doesn't get a score in such situations. Even the biggest maestros of football in history have missed them.
"One of the biggest stars in the world has not converted the penalty kick in the meantime," he added after Slovenia's Jan Oblak saved Cristiano Ronaldo's extra-time penalty on Monday.
Kek was dismissed from Slovenia's dug-out at the end of the first half of extra time when he was shown a red card, he said, for questioning the official about Ronaldo's penalty.
"I am sad we have to go back, I would to like to stay in this tournament," Kek added. "The match was in their hands, but we grew into this game."
Slovenia were solid throughout the tournament, which bodes well for the team's future, said midfielder Gnezda Cerin.
"I think we will be back at a major tournament before long and we will be able to live such moments again," he said.
"This is football. Portugal were perhaps a bit more focused at the shootout, but never mind. We've done a good job, not only for ourselves but for all of Slovenian sport. This team is something special and we'll be back."
