Qatari-bought fuel arrived at the Gaza Strip's only power station after entering through Israel on Tuesday in a bid to alleviate conditions in the Palestinian enclave, a Hamas spokesman and sources said.
The victim, Ari Fuld, 45, was well-known amongst settlers as a pro-Israel advocate. According to his Twitter account, he had planned a lecture tour in the United States in November.
The decision to open an investigation was a rare step and comes with Israel having faced international criticism over its use of live fire during unrest along the Gaza frontier.
A rise in cross-border rockets and air strikes in recent weeks has prompted the United Nations and Egypt to try to broker a truce to prevent another all-out conflict.
Israeli Army Radio said the rocket was a long-range Grad capable of reaching Israel's heartland and that it hit an open area outside Beersheba.
Gaza, under years of grinding Israeli and Egyptian sanctions aimed at isolating Hamas, has seen a surge in tensions since Palestinians launched weekly border protests on March 30.
Netanyahu will convene his security cabinet on Sunday to discuss an emerging deal, brokered by the United Nations and Egypt, for long-term peace in Gaza.