Thor is an Avenger for a multitude of reasons.
It's worth noting that the entire plot of The Avengers/Avengers Assemble is set into motion by Thor's actions in Thor. If Thor hadn't waged a war upon the Ice Giants out of pride and anger, then he never would have been banished to Earth, and Loki would have never sent the Destroyer there to kill him. Despite the fact that Thor saved the people of the small New Mexico town, they never would have been in danger if he hadn't been there, and he holds himself responsible for that.
Fast forward a year or so, and Loki is attacking the Earth with an army of Chitauri, and Thor heads to Earth in order to stop it. While confronting Loki in The Avengers/Avengers Assemble, he specifically claims that Loki is
Attacking the world I love for recompense for your imagined slights
believing that Loki's attack on Earth is a result of a personal vendetta Loki holds for Thor, and at the same time claims that
This planet is under my protection.
On top of that, he later learns that SHIELD are creating weapons of mass destruction after learning that there are other worlds out there. Nick Fury personally points his finger at Thor as the reason they are doing so, and Thor realises that, again, his interference with another planet has put its inhabitants at risk.
After saving New York, Nick Fury knows that his plan to create a response team, known as The Avengers Initiative, has been a success. Maria Hill queries how they can protect the Earth when they have all gone their separate ways (some of them pretty far), to which Nick Fury replies;
Because we'll need them to.
Nick Fury clearly believes that the next time the Earth is in danger, the team will reform to keep it safe.
At the end of the day, Thor's initial adventures on Earth, his love for Jane Foster, his responsibility in the Destroyer's attack on the New Mexico town, and the Chitauri Invasion of Earth have all meant that Thor has an interest in (or, as he may see it, a responsibility for) protecting the Earth. By the end of Thor: The Dark World, it would seem that Thor plans on spending more time on Earth with Jane Foster — which means that Thor truly is one of Earth's Mightiest Heroes.
It's also worth noting that the initial comic book line up of The Avengers included Thor, who lived on Earth at the time under the alias of Donald Blake. The tagline for The Avengers comics was "Earth's Mightiest Heroes", and Tony's line is a reference to this.