One thousand five hundred Italians, Americans and Italian-Americans crammed into the Italian Embassy yesterday evening to greet Italian President Sandro Pertini. A Little Italy sprang up on 16th Street as all those affiliated with Italian government, business and finance, the foreign service, the military and the media, were pressed together like grapes into wine.

"No president, no king, no head of state in the history of Italy has had such a relation with the man in the street," said Richard Gardner, former U.S. ambassador to Italy. "He is not just a figurehead, but has the scarcist of all resources--moral authority."

The mass of people moving in unison was so overwhelming that, at one point, the wooden floor of an enclosed garden gave way. There were gasps and "Oh's" as guests craned their necks to make sure the presidente was all right, but not much time to look as security men pushed Pertini on. "Did you feel that floor?" one Secret Service officer asked another after Pertini was safely inside a private study. "Yeah. I thought we were going all the way to the basement."

An outdoor pond or fountain had been camouflaged with wooden planks, then Astroturf, when a garden was covered by a huge tent to create an additional room. No one was hurt, but the weight caused the false floor to partially collapse beneath the anxious crowd.

Noone seemed to mind. "Mama mia!" could be heard throughout the swelling throng that pushed and laughed and clucked its thousand tongues, and "scusi-ed" and "prego-ed" and followed Pertini through the newly renovated rooms of the embassy, chasing him right into a den with a sturdy wooden door. Closed.

Only a small group of ambassadors and officials, including Attorney General William French Smith, Gardner and Italian Ambassador Rinaldo Petrignani, was able to enter the room and individually meet the Italian leader.

Pertini, 85, needed to rest, according to protocol official Mary Masserini, after he spent the day visiting Arlington National Cemetery, having lunch with Secretary of State Alexander M. Haig Jr., and speaking to a class at the Johns Hopkins School of Advanced International Studies.

Among those who waited hopefully at the door was Al Italia manager Fausto Bragnanti, who carried a copy of Pertini's book, "Sei Condanne Due Evasion," about the Italian resistance movement to the fascist regime. Bragnanti was finally allowed to enter the guarded room and get Pertini's autograph.

Asked why Pertini is so popular, Bragnanti said, "Are you aware that he's a Socialist? He's a man of very humble origins. He's the first president after the war that became close to his people. He's always running away from protocol and his people feel he's not a traditional politician."

Also waiting was Angela Cecere, who said of Pertini: "He's the most honest and spent all his life for Italian democracy."

After 40 minutes, the door was opened so Pertini could leave to dine at the ambassador's private residence. Once again, security fended off guests. But they still pursued the president, through the reception hall and down the front steps, cheering and waving at the dark windows of the limousine. Noted Gardner earlier: "Future historians will credit him with Italy's recovery."