The 19-year-old has made two appearances for the Gunners this season - against Coventry City in the League Cup in September and against Olympiakos in the Champions League in December.
"I had a great time in London and thank Arsenal for everything they've done," he told Nuremberg's official website. "Now I'm looking forward to the challenge of playing for FCN."
Nuremberg's general manager Martin Bader added: "Martin Angha has been brought to a top level of football at Arsenal over the last few years.
"He has shown his potential in London and will make the next steps in his development at FCN."
Angha is a versatile defender who can play anywhere across the back four but features mainly at right-back.