He said "the issue of capital punishment is for each and every member state to decide" and in conformity with international law.
The 62-year-old former South Korean foreign minister succeeded Kofi Annan of Ghana as the UN secretary-general on January 1. South Korea is among the 68 nations that retain the death penalty, although Seoul is considering abolishing it.
Meanwhile, Italy has begun a diplomatic push at the UN to put its drive for a worldwide moratorium on the death penalty on the international body's agenda, after criticism across Europe of Saddam's execution.
In New York, Italy's ambassador to the UN met the head of mission for Russia, which currently holds the Security Council presidency, to explain its determination to have the issue taken up by the General Assembly, the Italian foreign ministry said on Tuesday.
The office of Romano Prodi, the Italian prime minister, said that Italy would seek the support of other countries that oppose capital punishment.
On Sunday, Massimo D'Alema, the country's foreign minister, said that his country would work for the end of the death penalty worldwide.
Past lobbying by Italy for UN action against the death penalty has been unsuccessful. Italy is now one of the rotating members of the UN Security Council.
Groups such as Human Rights Watch have criticised the death penalty against Saddam, saying it was imposed after a "deeply flawed trial" with political interference.
Annan and leading UN rights officials also have opposed capital punishment, as has the European Union.
On Sunday, Ashraf Qazi Jehangir, the UN special representative in Iraq, had released a statement saying that the world body "remains opposed to capital punishment, even in the case of war crimes, crimes against humanity and genocide".
Ban's new spokeswoman, Michele Montas of Haiti, said: "The UN policy still remains that the organisation is not for capital punishment." She said Ban's comments were nuanced but would not say whether he agreed with Qazi.
On Tuesday in New York, Ban was greeted by a UN honour guard, went to a UN meditation chapel to honour fallen peacekeepers, spoke to reporters and held a mass meeting with UN staff.
Ban said the crisis in Sudan's Darfur region was "very high on my agenda" and he would meet Jan Eliasson of Sweden, his special envoy, on Wednesday morning.
He announced he would attend an African Union summit at the end of January in Addis Ababa, Ethiopia, and talk to Omar Hassan al-Bashir, Sudan's president.