The Court of Arbitration for Sport (CAS) rejected the appeal by Leon who had qualified by winning the CONCACAF Champions Cup in 2023.
"This is the greatest damage done to a team that has always competed with humility and honesty," Leon said in a statement.
"The only thing left for us is to win again to regain the place on the field that was taken from us. Have no doubt, as long as no one prevents us from doing so again, we will do it."
The expanded 32-team Club World Cup runs from June 14 to July 13 in the United States with $1 billion in prize money at stake.
Apart from the winners of each confederation's premier club competitions, teams also qualified based on their ranking based on their performances over a four-year period.
FIFA President Gianni Infantino has said the ruling body will look to organise a playoff between Los Angeles FC and Club America to decide which team will replace Leon in the tournament.