Israeli leaders have accused Damascus-based Hamas leader Khaled Meshaal of orchestrating the kidnapping, and have stepped up attacks in an effort to rescue 19-year-old Gilad Shalit.
Israeli Prime Minister Ehud Olmert said Israeli troops "would not hesitate to use extreme steps" in an effort to save the soldier.
Israeli troops launched an air strike on a Hamas training camp in southern Gaza.
According to witnesses, Israeli tanks staked out the old Gaza airport, with warplanes cutting out half of Gaza's electricity and pounding sonic booms over houses.
In response to the destruction to Gaza's infrastructure, Hamas said it kidnapped 18-year-old Israeli settler Eliahu Asheri in the West Bank. Militants said they would kill him if Israel did not halt operations in Gaza.
Also Wednesday, Israeli warplanes flew over one of Syrian President Bashar Assad's palaces to warn Syria against supporting Palestinian militants who abducted the soldier. The Israeli media said Assad was home at the time.
Olmert and the Israeli army have said that the increased assaults are to rescue Shalit and in no way an attempt to reoccupy Gaza.
"All the military activity of last night, which will continue in the next few days, is meant only to this end," Olmert said in a speech carried by Israeli television and radio. "We are determined to exhaust every possibility, so that we can realize the goal we have set ourselves."
On Tuesday, the Israeli attacks on Gaza pushed the feuding Palestinian factions, the Hamas-led government and rival Fatah Party, into drafting an agreement aimed at creating a unity national government.
If signed, the agreement would signal a major shift by Hamas, which for the first time would implicitly support a two-state solution to the Israeli-Palestinian conflict.
The Palestinian agreement comes at a time when Hamas is suffering from U.S. and European economic sanctions. The agreement would be a victory for Palestinian President Mahmoud Abbas, who had planned to put the two state solution before the Palestinian voters in a referendum in late July.