"Gaffer, it's a simple thank you," Robertson said in a lengthy Instagram post. "What you have done for our country will be remembered for years and years to come.
"In time, the sharpness of the disappointment we all feel right now will fade but not only will your legacy remain, hopefully it will become even more impactful."
Clarke, who led Scotland back to the World Cup after a 28-year-absence, resigned on Saturday after the team finished third in Group C but were not among the top eight third-placed finishers who progressed to the knockouts.
They opened with a 1-0 win over Haiti but then lost by the same scoreline to Morocco and were thumped 3-0 by Brazil.
"The moments that you helped deliver for our country, to name but a few - Serbia away, Spain at home, Norway away and of course Denmark at home. Games that will go down in Scottish folklore and outcomes that would not have been possible without you," Robertson, 32, said of previous qualifying results.
Scotland reached their first European Championship since 1996 after a penalty shootout win in Serbia took them to the 2020 finals and followed that up by making the Euro 2024 tournament after beating Spain 2-0 and winning 2-1 in Norway.
The Scots also beat Denmark 4-2 in their final group game in the 2026 World Cup qualifiers to book a place in North America.
Departing Liverpool defender Robertson, who will start the new Premier League season at Tottenham Hotspur, added: "26 boys lived their childhood dream but ultimately failed to achieve our goal."
He also thanked the Tartan Army, who are "a big reason Scottish people are loved and welcomed everywhere we go in the world."
FIFA World Cup 2026
The 2026 World Cup is taking place from June 11th to July 19th in the United States, Canada and Mexico. The tournament features 48 national teams and is played at 16 modern stadiums.
Match schedule and times | Group tables | How to watch the World Cup | World Cup Format | Past winners of the World Cup
