The Chile international, who made four appearances for his country at the World Cup, arrived at San Siro from south Wales on a four-year contract earlier in August following his former club's relegation from the Premier League.
Speaking at his official presentation on Monday, Medel was in no doubt that a move to Milan was the right choice.
"I am very happy to be at Inter," he told the press. "I spoke to [Genoa's Mauricio] Pinilla, who told me Inter are a big club.
"I'd like to thank everyone that brought me here; my team-mates have given me a warm welcome.
"There were other teams in for me but I always wanted a move to Inter and I'm here at last. I'm ready to play."
Medel was compared by the Italian media to former AC Milan midfielder Gennaro Gattuso but the 27-year-old maintains that he is very much in his own mould as a player.
"I'm Gary Medel and I want to show what I can do, perform well and win here at Inter," he added.
"I've already talked to Mazzarri but what was said will remain between us."
Inter kick off their Serie A campaign at Torino on Sunday but must first defend a 3-0 lead in the second leg of their Europa League play-off against Stjarnan on Thursday.