Nationals manager Dave Martinez called the reliever's actions "disrespectful to the organization."
"You're either in or you're in the way. I thought he was in the way," general manager Mike Rizzo said, adding the incident was "selfish."
Kelley was enjoying a strong season for Washington out of the bullpen. He compiled a 3.34 ERA and 0.96 WHIP with 32 strikeouts in 32.1 innings. His 35 pitching appearances were tied for the fourth-most on the team.
The 34-year-old Kentucky native struggled in 2017 en route to a 7.27 ERA in 33 games, but he finished with a sub-2.65 ERA in each of the previous two years. He owns a 3.73 career ERA across nine-plus seasons.
He previously played for the Seattle Mariners, New York Yankees and San Diego Padres.
The Nationals must decide whether to place Kelley on waivers, trade him or release him within the next seven days if they don't return him to the active 40-man roster. Since the non-waiver trade deadline passed Tuesday, he would need to clear waivers before Washington could deal him.
Sammy Solis and Justin Miller will take on more prominent roles in the Nats pen in front of closer Kelvin Herrera and setup man Ryan Madson.
Kelley should generate interest either via trade or free agency given his track record.