Atleti made the trip to Leicester with a 1-0 advantage from last week's first leg after Antoine Griezmann's controversial penalty.
Saul Niguez's first-half header left Leicester with a bigger mountain to climb and although Vardy provided hope of a comeback just after the hour mark, the Foxes ultimately came up short as Diego Simeone's men reached the final four with a 2-1 win on aggregate.
"We absolutely gave it our all, especially in the second half," England international Vardy told BT Sport.
"They made a lot of blocks on the line.
"We were in the ascendancy. We made a tactical change with Leo [Leonardo Ulloa] on to give an aerial threat. Unfortunately it wasn't to be. But we gave it our all and left it all out on the pitch.
"On another day some could have gone in. But we can be proud."
Noting Leicester were competing in the Champions League for the first time in the club's history, Vardy conceded his side were rank outsiders in the competition but still managed to outlast all of England's other representatives Arsenal, Tottenham and Manchester City.
"We knew it was going to be hard," he added. "Every game so far has been the same. From the display you've seen it was a fantastic effort.
"We have some unfinished business in the Premier League. That was a new experience for a lot of the lads and we have given it our all.
"The fans have got us through the 90 minutes."