Since its founding, Israel has seen itself as a modern-day Sparta, a tiny fortress nation-state in a hostile desert, whose survival depended on its internal cohesion and sheer military strength. All around it were Arab and Muslim enemies who denounced the Jewish state as a colonizing interloper, an outpost of foreign intruders who were bound to be evicted, sooner or later, like all their predecessors back to the Crusaders.

But Israel’s back-to-back agreements to normalize ties with the United Arab Emirates and Bahrain, to be marked in a signing ceremony at the White House on Tuesday — and the much-buzzed-about possibility that other Arab nations could follow suit — are causing some Israelis to ask whether a deeper shift may, after years in the making, finally be underway in the Middle East. Could their country at last be gaining acceptance in the region as a legitimate member of the neighborhood?

But Dr. Yitshak Kreiss, director general of Sheba Medical Center, Israel’s largest hospital, and a former military surgeon general, said that the biggest impact could be in changing the way ordinary Israelis think about their place in the region.

In an interview, Dr. Kreiss recalled setting up a field hospital in Haiti after the 2010 earthquake there. It could treat only a small fraction of the injured, but it had a great impact because “the most important thing for nations in a crisis is hope, the feeling that there’s a better future,” he said.

“When Israelis see the region opening, it doesn’t matter if it’s the Emirates or Bahrain, and next are Chad, or Oman, or Sudan, or Saudi Arabia,” Dr. Kreiss continued. “It’s the understanding that this can be a better region, that we should not accept things as they are. This is the strongest feeling Israelis can take from that.”