Following his country's exit at the Round of 16, the 30-year-old was granted an extended break before joining Antonio Conte's men a fortnight ago.
On Saturday, the versatile midfielder was introduced as a 71st-minute substitute for Dalbert Henrique as they settled for a share of the spoils against Valencia in their final pre-season game.
Mateo Politano's effort from the spot in the 82nd minute cancelled out Carlos Soler's 38th-minute opener for Valencia.
The Ghana international is pleased to have earned some playing minutes at the Mestalla Stadium.
"Happy for my first match of the season," Asamoah tweeted.
Asamoah will be hoping to play a part when Inter Milan begin their 2019-20 Serie A campaign with a fixture against newly-promoted Lecce on August 26.