The army said the missile was aimed at militants who were laying explosive devices in an area close to Israeli forces.
The attack comes amid a deepening crisis between Israel and the Palestinians over an Israeli soldier abducted by militants.
The armed wing of Hamas, which controls the Palestinian government, was among three militant groups to claim the capture of 19-year-old Gilad Shalit in a June 25 attack on an army post on the Gaza border, in which two Israeli soldiers and two militants were killed.
The captors have demanded the release of Palestinian prisoners and issued the deadline warning that after that time they would "consider the matter closed".
The deadline has already been rejected by Israel, which has sent troops and tanks into northern Gaza, after six nights of air attacks, the first time it has entered the territory since leaving last year after a 38-year occupation.
Israel has massed 5,000 troops at the Gaza border. "We will not conduct any negotiations on the release of prisoners," Prime Minister Ehud Olmert's office said.
Mr Olmert is holding the Palestinian Authority responsible for the safety of Corporal Shalit.
"Israel will not give in to extortion by the Palestinian Authority and the Hamas government, which are led by murderous terrorist organisations."
Chance for settlement
Mr Olmert also gave the green light to keep up military operations, after consulting with defence officials late on Monday, while Justice Minister Haim Ramon warned of even stronger repercussions.
"If, God forbid, they should hurt the soldier, our operations will be far, far worse," Mr Ramon told Israeli television. "The sky will fall on their heads if they dare to harm Gilad Shalit."
The Palestinian government said there is still a chance for a negotiated settlement.
"This problem must be resolved with wisdom and the language of reason," government spokesman Ghazi Hamad said.
"We believe there remains a chance to reach an acceptable formula. Since the soldier was abducted we have called for his life to be spared and for the problem to be resolved through diplomatic channels," he said in a statement.
The militant captors of the soldier are demanding Israel release "1,000 Palestinian, Arab and Muslim prisoners" as well as women and children held in Israel.
The three groups did not say what they would do, but warned that after that deadline they would "consider the matter closed and the enemy will be responsible for all results."
The statement was signed by the Popular Resistance Committees, the armed wing of the ruling Hamas movement and the Army of Islam.
The White House on Monday urged the militants to release the soldier, and said it is closely following events.