Born in Plymouth, Michigan, Gatt played varsity for four years at Detroit Catholic Central High School. He also played in the U.S. Soccer Development Academy for the Michigan Wolves. Gatt first attracted attention as a Michigan-area youth player when his club competed in a tournament against European academy teams in Switzerland, Belgium, Austria and Germany. Through the SuperElite system, the winger was offered trials at FSV Mainz 05 and at SCR Altach and eventually received an offer from the latter club.

Gatt proceeded to sign with SC Rheindorf Altach, leaving behind an opportunity to play at Indiana University. Upon his arrival in Austria in the summer of 2010, he immediately joined the first team, and scored 5 goals in 14 matches. On January 12, 2011 it was announced that Gatt had joined Norwegian side Molde. Gatt was called up to the United States National team on September 2, 2012 and made his international debut on November 14, 2012 in a friendly match versus Russia. Gatt left Molde at the end of 2016 and played with in the MLS on the Colorado Rapids. He rejoined Rheindorf Altach and after two years signed with Dundalk in Ireland. Josh moved back to the US in 2021 to join the USL Championship side Pittsburgh Riverhounds SC.

Josh joined the Michigan Wolves in 2022 to transition into coaching and personal training.