* Growing death toll in Gaza builds pressure on Netanyahu
* Israeli prime minister seeks quiet in Gaza, not control
* Confident Hamas could dictate Gaza endgame
By Dan Williams
JERUSALEM, July 21 (Reuters) - Shunning calls from political allies to "go all the way" against Gaza, Israeli Prime Minister Benjamin Netanyahu wants to avoid a protracted war that few see as winnable.
But deaths among Palestinian civilians and a mounting toll on Israel's ground troops could upend his strategy of a limited offensive aimed at ending Hamas rocket fire from the enclave, Israeli experts and foreign diplomats say.
Much depends on whether Israel can outrace waning understanding abroad for its sweeps of the densely populated fringes of Gaza cities and crush enough cross-border Hamas tunnels and rocket facilities before backing out.
It also depends on whether Islamist militants in Gaza are willing to accept any truce without wringing major concessions to free up the penned-in Palestinian enclave.
"As of now, the prime minister has not been swayed by all the clamour or the calls to go deeper into Gaza. He is not letting the background noise distract him," said Yaakov Amidror, a former national security adviser who served under Netanyahu.
"So the question that will be asked is: What next? That will depend on Hamas. If it keeps fighting, then we will too."
That could spell the sort of battlefield quagmire that has sucked in several of his predecessors.
More than 500 Palestinians, many of them civilians and including almost 100 children, have died since fighting flared on July. Almost 80 people died in the Shejaia district of Gaza on Sunday alone as Israeli forces battled Hamas gunmen.
Thirteen Israeli soldiers were killed in the same clashes -- the biggest loss of life on a single day for the army since 2006 and more than died during the last Gaza invasion in 2008-2009.
A grim-faced Netanyahu addressed the nation on Sunday, promising to remain steadfast, while sticking to his stated goal of restoring "quiet" to Israel and "significantly hitting the infrastructure of Hamas and the other terrorist groups in Gaza".
There was no suggestion that he wanted a renewed occupation of the coastal enclave, which Israel pulled out of in 2005, or the elimination of old foe Hamas.
"There's no guarantee of 100 percent success, but we will do our utmost to do the most we can," he said. Perhaps already seeking to frame a claim of victory, he added that military achievements had already surpassed expectations.
Despite growing international calls for an end to the fighting, Netanyahu has the backing of Israel's closest ally -- the United States, probably the only country in the world which could force Israel to back down in any fight.
A U.S. official briefed by Israeli commanders said the projected timeline for Israel's offensive had expanded after initial air and naval barrages failed to quell Hamas rocket fire and tunnel infiltrations.
The official, speaking on condition of anonymity, said that 'days' had stretched to 'a week or 10 days', accompanied by a recognition that waning international support meant Israel's time to achieve its goals was limited.
Underlining that point, U.S. President Barack Obama called Netanyahu on Sunday to express "serious concern" about the growing number of casualties in Gaza, the White House said, while his Secretary of State John Kerry headed to Egypt looking for ways to defuse the crisis.
At home, Netanyahu's standing has so far stood up well during the conflict. A poll for Knesset TV released on Monday said 71 percent of Israelis thought his handling of the crisis thus far had been good for the country.
Netanyahu is not a leader especially loved by his people, but he is seen as a safe pair of hands. For all his tough talk, the current operation is the first ground incursion he has ordered in his three terms in high office.
His perceived caution earnt him a public rebuke from a number of his ministers -- including from those belonging to his own Likud party -- who chastised him last week when he accepted an Egyptian ceasefire proposal before committing troops to Gaza.
Hamas rejected the deal and fighting progressed, but analysts said the unrest in his own camp during a time of conflict was unprecedented.
Foreign Minister Avigdor Lieberman, voicing the far-right view in the coalition, said he wanted Israel to "go all the way" and reoccupy Gaza. That would entail toppling Hamas and assuming control of 1.8 million impoverished Palestinians.
Clearly frustrated by all the sniping, and in a warning shot to Israeli hawks, Netanyahu last week dismissed Deputy Defence Minister Danny Danon for openly questioning him.
Unlike in previous flare-ups in Gaza in 2012 and 2008-2009, Israel has not faced any pressure to pull its punches from neighbouring Egypt -- one of only two Arab nations that Israel has made peace with and established diplomatic ties.
Having cracked down on its own Muslim Brotherhood, Hamas's ideological kin, Egypt feels little sympathy for the Palestinian group. But while that has itself contributed to the failed truce efforts, Cairo is uneasy about an open-ended Gaza crisis.
An Egyptian official briefed on the ceasefire talks said his government may eventually put the onus on Israel to de-escalate. The official argued that spiralling Palestinian civilian suffering was more damaging for Israel than for Hamas.
Netanyahu on Sunday accused Hamas of using images of "telegenically dead" Palestinians to further their cause, and more pictures like those that came out of Shejaia district on Sunday will certainly damage Israel's international image.
The fact Hamas rejected the initial ceasefire proposal has helped Israel portray itself as the wronged party, but such credit will not last for ever.
Amos Yadlin, a former chief of Israeli military intelligence who heads Tel Aviv University's INSS think-tank, said Netanyahu should benefit from setting "pretty modest" goals in Gaza.
"If the damage to the tunnels and the long-range rockets are internalised, and if hundreds of Hamas armed wing members are killed, that would constitute a hard blow and the prime minister could say the objectives were met," Yadlin said.
However, he said his leadership would be tested by any domestic misgivings over the growing number of slain soldiers.
"Whoever wages a war by looking at polls or TV ratings will not make the right decisions. Wars must be waged in accordance with strategic objectives," Yadlin said. (editing by Janet McBride)