Two first half goals from Giorgio de Arrascaeta, after Sergio Rochet saved Andre Ayew's penalty, looked like sending Uruguay through as Group H runners up with a 2-0 win over Ghana.
But Hwang Hee-chan's late goal gave South Korea a 2-1 victory over Portugal to secure second spot and dump Uruguay out on Friday.
Alonso said: "I don't have anything to say to my players because they broke their backs, they gave their best versions. I don't want to use excuses but everyone can see what happened in the previous matches.
"I know my players gave it all, they did their best to have a good match and today we can't say anything to our players.
"I believe in my players, they responded well, I am thankful. I deeply regret we are out of the World Cup, I truly believe in these players and what they could have achieved in this World Cup."
Former Inter Miami boss Alonso also declined to talk about his future after elimination.
"I think it's too early to talk about it (his future) let's wait for some days and we will discuss it," he said.
De Arrascaeta struck twice in six first-half minutes, heading in from close range before finishing off a flowing move with a low volley, after Ayew was denied from the spot.
The result for Uruguay was never in doubt after that but they could not find a third goal with Federico Valverde denied by Lawrence Ati-Zigi before former Liverpool and Sunderland defender Sebastian Coates poked wide late on.
Mohammed Kudus went close for Ghana, who knew a win would have sent them through, but the Black Stars are also knocked out.
Boss Otto Addo, talent coach at Borussia Dortmund, revealed he would step down having taken charge after Milovan Rajevac was sacked in January.
He said: "At the moment me and my family see our future in Germany, I always said if we qualified for the World Cup I would resign afterwards, even if we were world champions. This is my decision.
"Football is beautiful, sometimes it is ugly. It is ugly for us today, there is no human being in the world who hasn't missed a penalty.
"In the second half we had a lot of chances but we have to do better in defence, allowing two chances against these world-class strikers is two too much.
"We have to learn and I'm very sure we will learn from this in the future."