The World Group I tie against Norway will take place on September 16 and 17.
Nagal, who had beaten Uzbekistan's Denis Istomin in the opening round, went down 6-2, 6-1 to the second seed in just over an hour.
Nagal became only the third Indian to win a men's singles match at the Olympic Games.
Yuki Bhambri, with his protected ranking of 127, also made the cut but since he recently underwent a procedure on his right knee in the US, he has made himself unavailable.
The Haryana youngster crashed out of the Clay Court major with a 3-6, 3-6 loss to Alejandro Tabilo from Chile.
It was Nagal's best performance on the ATP tour after entering the event as a qualifier.
Nagal struggled to cope with the power-packed game of Lithuanian Ricardas Berankis.
Ankita's defeat means that Sumit Nagal will be the only Indian to compete in the singles event of the first Grand Slam of the season.
Prajnesh, who got the better of Turkey's Cem Ilkel in the opening round, lost 4-6, 6-7(4) to Vukic.
Thiem, seeking his maiden Grand Slam title, won an overwhelming 74 per cent of his first serve points and fired seven aces past the inexperienced Nagal, breaking him seven times in the two-hour contest.