Alcaraz, the defending champion at New York, exited this year's final major after suffering a defeat to third seed Daniil Medvedev of Russia on Friday.
Alcaraz will be replaced by Albert Ramos-Vinolas in the Spain squad.
"I was really looking forward to playing for Spain at the Davis Cup in Valencia, but I have to listen to my body after a very long tour," Alcaraz said on messaging platform X.
"I need to stop and rest, physically and mentally. The schedule is very demanding; there is still a long season ahead, and now it's time to recharge my batteries. Best of luck to the Spanish team! I will be supporting you strongly! Let's go!"
Ramos-Vinolas, currently ranked 88th, has been called by captain David Ferrer, who won the Davis Cup as a player on three occasions with Spain, lifting the trophy in 2008, 2009 and 2011, as well as a runners-up finish in 2012.
"Ramos was already called up for last year's Final eight in Malaga. This year he has been a finalist in Gstaad and a semi-finalist in Cordoba, as well as reaching the quarter-finals in Rio de Janeiro and the ATP Challenger in Parma," the country's tennis federation (RFET) said in a statement.
"In Davis Cup he has played five Davis Cup ties with a record of six wins and only two defeats."
Sixteen teams will compete in the group stage from Sept. 12-17 across Bologna, Manchester, Valencia and Split. The top two teams from each group will qualify for the knock-out stage from Nov. 21-26 in Malaga.
Spain will play Serbia, the Czech Republic and South Korea in Group C in Valencia.
