The UN chief earlier condemned such attacks as "unacceptable," Ban's spokesman Martin Nesirky said.
"The secretary-general is deeply concerned and calls for an immediate cessation of acts of terrorism and violence against civilians in order to prevent further escalation and loss of life," he said.
But the Israeli envoy to the United Nations, Meron Reuben, sent a letter to Ban and the UN Security Council suggesting that his condemnation of the bomb attack -- the first in Jerusalem in six years -- did not go far enough.
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"In view of these troubling developments, Israel expects the Security Council, the secretary-general, and the international community to condemn all of these attacks in very clear terms," Reuben said in his letter.
Ban and the UN should "send a firm message to these terrorists and their patrons, which seek to escalate conflict," he added.
At least one person was killed and more than 30 were wounded when a bomb ripped through a Jerusalem bus, just hours after militants vowed revenge for two deadly Israeli raids on Gaza.
Police and officials said the bomb had been hidden in a bag at a bus stop near Jerusalem's central bus station at the western entrance to the Holy City.
"The Security Council and the international community must also devote much more attention to preventing the smuggling of arms into the Gaza Strip," Reuben said, adding this issue "does not receive the appropriate attention that it deserves."
