Since its founding, Hamas has declared that Israel has no right to exist, that there are no Israeli civilians and that every Israeli citizen is a soldier of the state, and thus a legitimate target.

Still, if Western nations considered Hamas a terrorist organization, they also thought that it was preoccupied with governing Palestinians crammed into Gaza. Hamas provided social services. It was even thought of as a restraint on what were considered even more radical groups.

In Israel, successive governments cut quiet deals with Hamas, hoping to keep a form of stability in the Gaza Strip, which the group controls, especially after the Israelis withdrew unilaterally from the territory in 2005.

But the assault launched by Hamas this weekend, with more than 900 Israelis listed as killed so far and more than 150 believed taken as hostages and human shields into Gaza, has now stripped away any remaining illusions about the group or its intentions. The attack by Hamas into Israel proper is notable for its terror, targeting not only uniformed soldiers, but also civilians, including women and children.

Senior Israeli officials now say Hamas must be crushed, both to restore stability in Gaza and credibility for Israel as an ineradicable part of the Middle East.

“We must admit that the conception was wrong, we can’t hide behind it,” said Tamir Hayman, a retired major general and managing director of Israel’s Institute for National Security Studies.

There is much the same disillusionment in the West, especially among Europeans who have provided significant aid to Gaza, some of which has always been siphoned off by Hamas. The horrors of the weekend now cast Hamas in a new light, one which is likely to have a major effect on events going forward.

The European Union, like the United States, has labeled Hamas a terrorist organization and officially boycotts it, but many Europeans see the group as freedom fighters struggling against an Israel that is slowly making a Palestinian state impossible.

For many in the West, especially the young and those on the left, “Gaza is a one-word argument for Israel’s brutality toward a blockaded enclave living in miserable conditions,” said Natan Sachs, director of the Center for Middle East Policy of the Brookings Institution.

Hamas, for them, was “fundamentally a nationalist resistance movement in the context of Gaza.” That view was shattered “for some, if not all, on Saturday,” he said.

In Europe, there have been uniform official condemnations of the attacks and of support for Israel. But tellingly on Monday there was confusion in Brussels, when an E.U. official, Oliver Varhelyi, announced that 691 million euros, or about $730 million, in aid to the Palestinians would be put under review, an announcement quickly softened to say that humanitarian aid would continue.

In Israel, the military had few illusions about Hamas, considering it among the most extreme of the Palestinian armed groups and recognizing that it would never accept any form of recognition of Israel, unlike Fatah, the heart of the Palestinian Authority, Mr. Hayman said in an interview.

The Authority, set up after the Oslo accords of the 1990s, controls the West Bank, and Israel has tried to strengthen it while working with the Authority to weaken Hamas in the West Bank.

Yet for Israeli leaders, Hamas was useful, too. It was someone in control of Gaza to talk to, Mr. Hayman said, that could help keep stability, which is why Israel had refrained from a full-scale assault in Gaza, he said. “This conception has failed.”

Yaakov Amidror, a retired major general who served as national security adviser to Prime Minister Benjamin Netanyahu in an earlier government, agreed.

“It’s a huge mistake that I did, believing that a terror organization can change its DNA,” he said. “I thought that Hamas, because of its responsibility and because it’s not only a terror organization, but also an organization with ideas about the future, a small branch of the Muslim Brotherhood, is more responsible, and I learned in the hard way that it is not so, that a terror organization is a terror organization.”

Mr. Amidror, now a senior fellow at the Jerusalem Institute for Strategic Studies, said bluntly: “We don’t want to make the same mistake again.” Hamas, he said, “should be killed and destroyed.”

Mr. Hayman also foresees a strong, prolonged Israeli response. “The context right now is after a brutal, unyielding terror activity of a kind Israel has never seen, worse than the atrocities of ISIS, with the slaughtering of people, the torturing of women and abducting children and old people,” he said. “This is a kind of madness which we never imagined.”

The Israeli military has launched a number of retaliatory strikes into Gaza since Saturday morning. Already, more than 680 Palestinians in Gaza have been killed in Israeli strikes, Gazan officials say.

Israel did not see “the strategic meaning of the rhetoric of Hamas,” said Shlomo Avineri, a former Israeli official and a political scientist. “It was dismissed as rhetoric, without considering how vulnerable Israel is, with all the kibbutzim near Gaza.”

When Hamas said that “every civilian is a soldier, this was not rhetoric but identifying the vulnerability of the Israeli communities inside Israel,” he said. Instead, he said, the army was focused on individual terrorism in the West Bank, and the government on its controversial efforts at judicial reform.

For many Palestinians, Hamas was a military organization using the only means it had to resist a far superior Israeli military and Israeli occupation, including terrorist acts, suicide bombings and rocket attacks.

For Israelis, Hamas’s brutality was clear from the suicide bombing campaign of the 1990s and early 2000s, and “Gaza is a one-word argument for the danger of unilateral withdrawal and trusting in Palestinian rule,” Mr. Sachs said.

Much will now depend on how the international community reacts to the inevitable deaths of civilians in Gaza, a tightly packed space where Hamas has had time to prepare its defenses.

Much will also depend on whether Hezbollah joins the fight from Lebanon, as it did in 2006. After Israel’s cautious effort to hurt Hezbollah then, few expect many limits this time on either side. With Hezbollah’s leader, Hassan Nasrallah, reportedly regretting his attack on Israel in 2006, but also equipped by Iran with far more sophisticated rockets, mutual deterrence may yet win out.

But if Mr. Amidror is correct, Israel will pursue its war against Hamas with little regard for Western opinion and criticism. “It is the last time that we can allow Hamas to be strong enough to attack Israel,” he said.

If Israel succeeds in destroying Hamas, he said, “it will show that when there is a real test, Israel is ready to pay the price, ready to fight and ready to make the difference. I think that we will be more appreciated by everyone in the Mideast, not just the Saudis. If the reaction of Israel will not be strong enough, we might lose some support in the Middle East.”