The American has been the team's Formula One reserve driver for the past two seasons, and has driven for the team in three Grand Prix weekend practice sessions, including an outing on home soil at the Circuit of the Americas last year.
The 22-year-old has also raced for Caterham's GP2 affiliate for the past two seasons, and before that was part of their Formula Renault 3.5 programme in 2012.
"I will no longer be racing for Caterham as of today," Rossi said.
"It's been an emotional couple of years with the team, the highlight being driving with them in Austin last year. Now it's time to move on."
Rossi says he remains focused on securing a seat in F1, adding: "My goal is to be in F1 and add value to the sport.
"I'm very optimistic about my next steps; we have a great opportunity to grow the sport in the States and I believe the timing is perfect."
Rossi's departure comes after Caterham announced a management takeover at the start of the month, with Tony Fernandes selling the team to a consortium of Swiss and Middle Eastern investors advised by former HRT and Midland team principal Colin Kolles.
Rossi subsequently announced that he will drive for Campos in Germany this weekend in GP2.