Israel, which is fighting Iran-backed Hezbollah in Lebanon, promised to retaliate, stoking fears of a wider war.
Carolineskolen, a Jewish school located near the embassy in the Danish capital, would stay closed on Wednesday due to its proximity to the crime scene, a spokesperson for the Jewish Community in Denmark told Reuters.
Several other embassies and consulates are also located in the area.
There have also been several recent security incidents near Israel's embassy in neighbouring Sweden, where police on Tuesday said they were investigating suspected gunfire in the area.
In January a Stockholm police bomb squad disarmed what investigators called a "dangerous object" outside the Israeli embassy building.
The incidents in Sweden caused no injuries or significant damage.
Swedish authorities have said security police averted several planned attacks linked to Iranian security services using local criminal networks. Iran has called the Swedish report "baseless"