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India lost the tie 2-3 on Saturday with Holger Rune starring for Denmark.
Felix Auger-Aliassime beat Alex de Minaur 6-3, 6-4 to give them an unassailable 2-0 lead over Australia in the final.
Casper Ruud combined with Viktor Durasovic to win the doubles match wrap up the tie.
Kumar as a non-playing captain played a key role in including a 16-year-old Leander Paes in the Davis Cup team against Japan in 1990.
Zverev has not played since June after undergoing surgery to fix damaged ankle ligaments following an injury he suffered during his French Open semifinal.
The veteran doubles ace has opted out of the away tie due to a knee injury.
The World Group I tie against Norway will take place on September 16 and 17.
The away tie will clash with the Asian Games in September.
Indian sports arena will see many champion performers in the years to come. But it is unlikely that one will see a fighter in the same mould as Paes.
Rohan Bopanna and Divij Sharan saved three match points in an edge-of-the-seat thriller to prevail over the Danish pair of Frederik Nielsen and Mikael Torpegaard.