Stäbler represented Germany at the 2012 Summer Olympics in London, where he competed in the 66 kg class in men's Greco-Roman wrestling. He received a bye for the second preliminary match, before losing out to Hungary's Tamás Lőrincz, who was able to score six points in two straight periods, leaving Stäbler with a single point. Because his opponent advanced further into the final match, Stäbler offered another shot for the bronze medal by successfully defeating United States' Justin Lester in the repechage bout. However, he lost the bronze medal match to Georgia's Manuchar Tskhadaia, who pushed him out of the wrestling mat in the third period, with a score of 1–3.