* Speech follows Israeli air strikes on Gaza
* Egypt's Israel ties have cooled since Mubarak
* Israel relations present Mursi with dilemma
By Marwa Awad and Omar Fahmy
CAIRO, Nov 15 (Reuters) - Egyptian President Mohamed Mursi condemned Israeli air strikes on the Gaza Strip as unacceptable on Thursday, in his harshest public criticism of Egypt's neighbour since taking office in June.
Mursi's televised address came as a stand-off deepened between Israel and militant groups in Gaza. A Hamas rocket killed three Israelis north of the Strip and the Palestinian death toll from Israeli attacks rose to 13.
"We are in contact with the people of Gaza and with Palestinians and we stand by them until we stop the aggression," Mursi said. "The Israelis must realise that this aggression is unacceptable and would only lead to instability in the region".
Looking more subdued and downcast than in previous speeches, Mursi seemed ill at ease as he listed steps he had taken to recall Egypt's ambassador in Israel and appeal to the United Nations Security Council.
It was the first time he mentioned Israel by name in a public address. Ties between the two neighbours were never warm but have cooled further since Mursi's predecessor Hosni Mubarak, a staunch U.S. ally, was deposed in a street revolt last year.
Mursi faces an uncomfortable dilemma in his relations with Israel. While keen to acknowledge widespread popular antipathy to the Jewish state, he also needs to persuade Western powers that they need not fear an Egypt governed by Islamists.
Israel is wary of the rise of the Muslim Brotherhood, the group that won most seats in post-Mubarak elections and which propelled Mursi to power.
The Brotherhood, which also inspired Hamas, describes Israel as a racist and expansionist state, although Mursi has pledged to respect a three decade-old peace treaty that ended a succession of wars with Israel.
The head of the Brotherhood, the country's most organised group, called for nationwide protests to support the people of Gaza. Demonstrations were also planned for Friday.
Dozens of youths protested in front of the Arab League headquarters in Cairo and burned Israeli flags, chanting: "We will not give in, no matter how much the brutality grows". Protesters in the port city of Alexandria also burned Israel flags.
Mursi said he had spoken by phone with U.S. President Barack Obama and discussed "ways to reach calm and end the aggression".
He said he told Obama of "how keen we are (to maintain) relations with the United States but also our absolute rejection of this aggression and the spilling of blood and the blockade of Palestinians".
He said U.N. Secretary General Ban Ki-moon had promised to relay his demand for an end to the violence to the Israelis.