The Spain international was sanctioned for running down the sidelines to celebrate Gareth Bale's goal in Los Blancos' 4-1 win over Atletico Madrid in the Champions League final in May.
Madrid challenged the ruling but the Uefa Appeals Body has upheld the original decision, meaning Alonso will sit out Tuesday's Super Cup clash with Europa League winners Sevilla.
The midfielder had been forced to sit out Madrid's meeting with Atleti in Lisbon after picking up a booking in his club's semi-final second-leg win over Bayern Munich at the Allianz Arena.
Consequently, he had been consigned to the touchline for the final. However, Alonso set off down the touchline in celebration when Bale put Madrid 2-1 up in extra-time, resulting in a one-game ban for the 32-year-old.
Los Blancos were also fined £14,000 for picking up five bookings in the final, as well as the "improper conduct" of their supporters, who were found to have set off fireworks during the tournament decider.