Leon recorded his first big league hit in the seventh inning of Washington's 9-5 loss to the Mets, finishing the day 1-for-3.
Solano was placed on the 15-day disabled list, retroactive to July 18.
Jesus Flores has been taking a heavy physical toll as the team's everyday catcher while also seeing his batting average drop to .230. Solano had thrived in his first Major League action, batting .314 with two home runs, six RBIs and a .923 OPS in 35 at-bats since his debut on May 29.
"We like both of our backup catchers," general manger Mike Rizzo said. "Solano did a great job for us and Leon is a young player, but he gives [manager] Davey [Johnson] some versatility as a switch-hitter off the bench, and he's a catch-throw guy that was playing extremely well in Double-A when we called him."
Leon also saw his first action in the big leagues earlier this season, but lasted just four innings after suffering a high right ankle sprain during a collision at home plate against the Padres on May 14. The 23-year-old Venezuelan catcher hit .313 with 15 extra-base hits and 22 RBIs in 45 games with Harrisburg and Class A Auburn.
"It was pretty tough, because that's the dream for all players, to play in the big leagues," Leon said of his injury, adding that it took him about two weeks to feel confident playing on it. "But it happens, and I'm back, that's the good thing."