The Centre held that seniority is not the sole criterion for selection.
Verma also sought quashing of a note issued by the defence ministry rejecting his plea against Singh's appointment.
Vice Admiral Singh is scheduled to take charge as the new Navy chief from incumbent Admiral Sunil Lanba, who is set to demit office on May 31.
This will be the first instance since Independence that two of the three service chiefs will be the government's chosen men.
In his fresh petition, Verma said he had filed a statutory complaint with the defence ministry, challenging appointment of Vice Admiral Singh as the next navy chief, and asked it to respond within 10 days.
In his petition, the vice admiral has challenged the government's move to ignore seniority and appoint his junior as the next chief of navy.
Vice Admiral Nirmal Kumar Verma, who once commanded India's sole aircraft carrier INS Viraat, on Monday took over as the new Navy chief.
"Targeting ships at high seas is not an easy task or as simple as land warfare," Admiral Verma told media persons on the sidelines of a seminar on aircraft carriers organised by the National Maritime Foundation in New Delhi.
As Navy chief, Admiral Singh's immediate priority will be to expedite the long-delayed modernisation of the Indian Navy, including by inducting new warships, submarines and aircraft.
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