Bhupathi and his Austrian partner lost 4-6 3-6 in the first round, which lasted only 67 minutes.
Bhupathi continues to struggle in his comeback as he returned to on court action after a nine-month break during which he set up IPTL.
He played his first tournament at Chennai Open where he had lost in the second round with compatriot Saketh Myneni.
Bhupathi and Melzer committed a staggering 27 unforced errors to just nine of their opponents, who served extremely well.
The 6 feet 2 inch Zeballos was serving bombs and his service made a big difference to the outcome of the match.
On Melzer's serve, the Austrian and Bhupathi saved the first two but Melzer sent a forehand to net on the third chance and threw his raquet in frustration.
In an ominous start to the second set, Bhupathi was broken in the second game. Schwartzman hit a service winner to make it deuce and the Indian only helped the rivals by double faulting, which put them down by a break point.
Schwartzman served solid to make it 3-0. Bhupathi and Melzer had a chance to get that break back on Zeballos' serve but the tall southpaw yet again came up with big serves to save both. In just 19 minutes they had a commanding 4-1 lead.
Bhupathi and Melzer held their serves after that but the lead stayed with the Argentines and they easily pocketed the set and match.
Bhupathi's forehand went wide to hand the Argentines two match points and they converted first when a double-handed backhand by the Indian kissed the net.
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