Veteran Albert Ramos-Vinolas (35) will replace him for the matches in Valencia between September 12th-17th, said the Spanish tennis federation in a statement.
Alcaraz had been set to lead Spain and clash with rival Novak Djokovic (36), named in Serbia's team, whom he beat in a thrilling Wimbledon final in July.
"I was very excited to play the Davis Cup for Spain in Valencia, but I have to listen to my body after a very long tour," wrote Alcaraz on X, the social media platform formerly known as Twitter.
"I need to stop and rest, physically and mentally. The calendar is very demanding, there's still a lot of the season left and now I have to pick up my strength."
Alcaraz's defence of his US Open title was ended by Medvedev who came out on top in their semi-final 7-6, 6-1, 3-6, 6-3 on Friday.
Spain will face Serbia, the Czech Republic and South Korea in their Davis Cup group, aiming to qualify for the last eight, played between November 21st-26th in Malaga. The top two will progress.
