The 18-year-old midfielder has penned a five-year contract with the Bundesliga outfit, opting to remain in his homeland instead of negotiating a lucrative move to the Premier League.
Gunners boss Arsene Wenger is understood to have been monitoring Goretzka for well over a year, and wanted to bring the Germany U19 international to the Emirates Stadium ahead of the 2013/14 campaign.
New Manchester United manager David Moyes has also been credited with an interest in the talented teenager, but missed the opportunity to lure the player to Old Trafford after Schalke secured Goretzka's signature.
"In the form of Leon Goretzka we're delighted that, alongside Max Meyer, another incredible young talent has decided to ply his trade with FC Schalke 04," enthused manager Horst Heldt after completing the signing.
"Leon belongs among the most talented players in the whole of Germany and has a huge amount of potential.
"There were several clubs interested in acquiring his services, but believes he has the best opportunity to develop himself further as an athlete with us."
Manchester United and Arsenal will now be forced to turn their attention to alternative targets this summer, as they look to strengthen their respective squads before the close of the transfer window on September 2.