Touted as dark horses before the tournament, Austria failed to reproduce their impressive qualifying form, losing 2-1 to Iceland in their final group game last week to seal their fate.
"The match against Iceland was my last one in red-white-red," said Fuchs, 30, in a video message posted on his Facebook page.
"I'm very proud of the 10 years I spent with the national team, I did everything with passion," the former Schalke player added.
The left back signed a three-year contract last summer with Leicester and played a key role in their shock run to the Premier League title.
A full international since 2006, Fuchs earned 78 caps for his country and scored one goal.