Mayor Bloomberg has in the past recounted a childhood memory of his father, a bookkeeper who made $11,000 a year, stretching thin his bank account to write out a $25 donation to the NAACP.

Yesterday, Mr. Bloomberg was in Israel to give away a portion of his personal fortune, some in his late father’s name. The mayor, a billionaire who made his money with his financial information company, traveled to Jerusalem to announce the renovation and expansion of Jerusalem’s flagship ambulance center, which will be renamed the William H. Bloomberg Magen David Adom Jerusalem Station.

“My father taught me the importance of giving back, and I’ve tried to follow his example and give what I can to those organization working for the good of all,” Mr. Bloomberg, who had his 98-year-old mother, his sister, and one of his daughters with him, said.

The mayor, who is staying in the country through Friday before traveling to Jordan, met with Israel’s prime minister, Ehud Olmert, and with the acting president, Dalia Itzik.

He also met with the families of three Israeli soldiers who were kidnapped by terrorists over the summer, and demanded that the men be released without conditions. The kidnappings, which happened in June and July, sparked a standoff between Israel and the Palestian Arab group Hamas that led to the war in southern Lebanon.

Mr. Bloomberg ‘s backing of a United Nations resolution for the soldiers’ release and a trip he’s taking Friday to the Israeli city of Sderot — which borders Gaza and has been frequently bombed with rockets by Hamas — fortify the already strong relationship he’s forged with the country. He is also traveling to Jordan, where he will meet with King Abdullah II, a longtime American ally, and with the American ambassador to Jordan.

The dedication of the new medical facility also highlights Mr. Bloomberg’s devotion to the same public health causes he has championed in New York. The new facility means the Bloomberg moniker will be plastered on yet another building in the city at a time when Mr. Bloomberg is said to be weighing a 2008 run for the White House.

In 2003, Mr. Bloomberg and his family dedicated a medical facility in Jerusalem to his mother. The Charlotte R. Bloomberg Mother and Child Center is a freestanding clinic at Jerusalem’s largest hospital, the Hadassah University Medical Center.

Yesterday, the mayor said his mother stopped in Lisbon on her way to Israel. She apparently said she didn’t know when she would get to go there and wanted to see it now.

Mr. Bloomberg’s father died in 1963, when the mayor was a 21-year-old student at Johns Hopkins. Yesterday, Mr. Bloomberg recalled his father sitting in the family living room in Massachusetts, smoking a pipe and talking about world affairs.

When asked about his presidential ambitions, Mr. Bloomberg demurred, joking that Magen David Adom already has a president and doesn’t need another one. He said he doesn’t “plan” to create a presidential exploratory committee, as other candidates have done.

Mr. Bloomberg called Israel the “home of democracy” and defended the country against Hamas and Hezbollah. He said Israel should not recognize Hamas unless the group recognizes Israel’s right to exist and until it denounces terrorism. He also said it would be “very bad” if Iran becomes a nuclear power.

The mayor is traveling to Ireland on Sunday to announce the addition of direct airline flights between Ireland West Knock and John F. Kennedy International Airport.