Hamas called on Palestinians to "resist" the march.
Hamas and other militant groups in Gaza did not immediately respond with rocket fire into Israel, but the situation along the Israel-Gaza border remains extremely tense, and the possibility of an imminent and serious escalation can not be ruled out.
The counterstrike, which happened Tuesday evening in the US and early Wednesday morning local time, targeted Hamas military compounds, according to the IDF.
The strikes, the military said, came in response to the launching of the incendiary balloons, which the Israeli fire brigade reported caused 20 blazes in open fields in communities near the Gaza border.
While the route of the march usually does pass through the Damascus Gate, police have ordered that it be changed so that marchers enter the Old City through the Jaffa Gate instead, passing and pausing outside of the Damascus Gate on their way there.
It followed a march by Jewish nationalists in occupied East Jerusalem on Tuesday, which had drawn threats from Hamas, the militant group that runs Gaza.
Palestinians consider the annual march, meant to celebrate Israel's capture of east Jerusalem in 1967, to be a provocation.
From the Damascus Gate, they proceeded around the Old City to the Western Wall, the holiest place where Jews can pray.
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The parade, scheduled for Tuesday, creates an early test for the fledgling government led by Prime Minister Naftali Bennett - a patchwork of parties that includes hard-line nationalists as well as the first Arab party to sit in a governing coalition.
Police said that officers arrested 17 people suspected of involvement in violence, some of whom threw rocks and attacked police, and that two police officers needed medical treatment.
"The right to demonstrate is a right in all democracies", said Internal Security Minister Omer Bar-Lev.
At the last minute, that march was diverted away from the Damascus Gate and the Old City's Muslim Quarter, but the move was not enough to dissuade Hamas from firing rockets toward Jerusalem, attacks that set off last month's round of fighting.
JERUSALEM, ISRAEL - JUNE 15: Police officers clash with Palestinian as they force Palestinians out of Damascus Gate area before the far right flag march on June 15, 2021 in Jerusalem, Israel. Palestinian protesters repeatedly clashed with Israeli police over restrictions on public gatherings during the Muslim holy month of Ramadan in April and May.
The conflict between the two sides between May 10 to 21 had led to the deaths of at least 260 Palestinians.
Israel's annexation of east Jerusalem is not recognized by most of the global community which says the city's final status should negotiated with the Palestinians - who claim the city's east as the capital of their future state.
Defence Minister Benny Gantz met with the military chief of staff, the police commissioner and other senior security officials.