Hamas has been leading the protests to ease a crippling Israeli-Egyptian blockade that was imposed in 2007 in order to weaken the militant group.
"Two of them were intercepted by the iron dome", he said of Israeli air-defense system.
Rocket-warning sirens were later sounded in southern Israel.
More than 200 Palestinians have been killed and thousands injured by Israeli forces during the Gaza protests in the past seven months, according to the enclave's health ministry.
Arab media reported that at least six senior Hamas commanders were killed and more were injured in the raid by IDF commandos. He was identified as Nour Baraka.
In the West Bank, Abbas angrily accused Israel and the United States of ganging up behind his back to entrench Hamas's control over Gaza.
The Israeli army said its forces carried out a "security operation" in the Gaza Strip, followed by violent clashes. The operation had ended and the families of the soldiers were notified, it added.
Israeli critics of the transfer say it will benefit Hamas, with some dismissing it as extortion by the militant group, which is sworn to Israel's destruction.
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An Israeli ground operation to kill or abduct militants inside the Gaza Strip would be rare. Israeli Defence Minister Avigdor Lieberman was holding consultations at military headquarters, his office said in a statement cited by Reuters.
Netanyahu had earlier defended his decision to allow Qatar to transfer the cash to Gaza despite criticism from within his own government, saying he wanted to avoid a war if it wasn't necessary.
Israeli military spokesman Lt. Col. Jonathan Conricus said the operation was "not meant to kill or abduct terrorists but to strengthen Israeli security".
Naftali Bennet, Netanyahu's education minister and right-wing rival, compared the cash flow to "protection money" paid to criminals.
"Everyone knows where that money is going to go ultimately", said Netanyahu's agriculture minister, Uri Ariel.
Violence has flared frequently on the frontier since Palestinians began weekly protests on March 30.
A gun battle erupted and Israeli tanks and aircraft opened fire in the area, witnesses said.
Over 170 Palestinians have been killed by Israeli fire in the border protests, in which Palestinians throw rocks, burning tyres and grenades toward Israeli troops.
On Friday, Palestinian civil servants began receiving payments after months of sporadic salary disbursements in cash-strapped Gaza, with $15 million delivered into the enclave through Israel in suitcases.