Addo was named the Black Stars' temporary manager in February, although he has been balancing the position with his work as a talent coach for Bundesliga powerhouse Borussia Dortmund.
With a victory over Uruguay on Friday at Al Janoub Stadium, Ghana would have advanced to the round of 16, but instead, the African nation placed last in Group H after a 2-0 loss.
Giorgian de Arrascaeta scored twice quickly after Andre Ayew missed a penalty in the first half to give Uruguay the lead. However, the victory was insufficient to advance Diego Alonso's team to the knockout phase.
Addo made his resignation known during the news conference that followed the game.
"It was obvious I would stop after the World Cup when I joined as an assistant coach in October last year," he stated.
"At the moment, my family and I envision our future in Germany; we are content in Dortmund, where I like my job.
"Even if we were world winners, I would still be going; this is a family decision. This choice is mine.
The youngest team in the competition was Ghana, and Addo believes the team has a promising future.
It wasn't a great day for us, but that's football. Sometimes it's beautiful, and other times it's absolutely terrible.
No player has ever missed a penalty, and their goalkeeper was excellent tonight.
We had plenty of opportunities, but our defence needed to improve. Giving world-class strikers two chances is excessive. We have a youthful team, which will need to learn but is undoubtedly going places.
"We need to have confidence in ourselves. The best athletes excel because they had previous success. That results from making mistakes.
They must work on each of these issues. They will learn from this and improve for the future.