Meanwhile, Sherry Rehman, leader of Pakistan's main opposition party PPP, said Pakistan's case against India at ICJ was "weak".
Jadhav, a former navy officer, was sentenced to death by a Pakistani military court last month after being convicted of spying.
A Pakistani lawyer added that a bilateral agreement allows either country to make decisions on consular access in cases involving politics or security.
The United Nations' highest court on Thursday prevented Pakistan from executing a former Indian naval officer, pending a final decision from the 11-judge bench later this year.
The court has asked Pakistan to give consular access to Jadhav, who was arrested in March a year ago in the country's southern province, a few kilometers (miles) from the Iranian border.
He said Pakistan's security is very important to them and they have to maintain it as their fundamental sovereign right.
Jadhav was reportedly arrested from Pakistan's Balochistan province on March 3, 2016.
India has maintained that Mr Jadhav had long retired from the navy, and that he was kidnapped from Iran while there on personal business, and smuggled into Pakistan. "But (Jadhav) can not be executed until the stay order is there", Usmani said.
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"The decision to stay Jadhav's execution pending ICJ hearing is unanimous", said ICJ President Ronny Abraham.
Pakistan Foreign Office spokesperson Nafees Zakaria on Thursday said the International Court of Justice (ICJ) has no jurisdiction to hear the case of alleged Indian spy Kulbhushan Jadhav.
India has appealed to the world court that Pakistan did not heed an worldwide treaty that granted diplomatic immunity to foreign nationals charged with capital crimes. The argument was rejected by the ICJ.
Though India and Pakistan are signatories to the Vienna Convention since 1977, New Delhi's request for access to Jadhav was denied 16 times.
Kulbhushan Jadva's friends expressed satisfaction and joy after the International Court of Justice (ICJ) stayed his execution by a Pakistan military court.
He also ordered Pakistan to inform the tribunal that it has implemented the decision, and stressed that ICJ decisions are binding on all member states.
Harish Salve was also part of the team that successfully argued at the International Court of Justice (ICJ) against a bid by Marshall Islands to sue India for violating the nuclear disarmament obligations.
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