He was given a short tour of the museum before meeting two survivors of the Holocaust, in which six million Jews were murdered during World War II. It is situated on the western slope of Mount Herzi, overlooking the city of Jerusalem.
According to Israeli website Ynet, Mr Elki described it as "regrettable" Britain had chosen to politicise the royal visit.
Though the trip is being billed as non-political, and places a special emphasis on technology and joint Israeli-Arab projects, the prince is meeting with Israeli and Palestinian leaders and visiting sites at the heart of the century-old conflict.
"Need to come back with the family for this shot", he said.
Wearing a black Jewish skullcap, William, second in line to the throne, laid a wreath in the Hall of Remembrance at the Yad Vashem memorial, where an eternal flame flickers and the names of extermination and concentration camps are engraved in the floor.
Later in the day, William planned to meet young people participating in a soccer-based youth programme in Tel Aviv and deliver a speech at a reception at the British ambassador's residence.
He is also due to meet the Palestinian Authority president, Mahmoud Abbas, in Ramallah in the West Bank.
Fans of the royal family were quick to reply to the photo, with one writing: "The Queen must be very proud of Prince William. And he really wants to get under the skin of the country", Quarrey added. He is expected to tour the Western Wall, Muslim and Christian holy sites, and occupied Palestinian territories.
Israel captured east Jerusalem from Jordan in the 1967 Mideast war and annexed it in a move not internationally recognized.
The visit comes just after Israel marked its 70th anniversary of independence and amid surges of violence along the Gaza border.
However, The Sun reported William said she loved her time living in Jordan as a child and commented, "we'll have to bring the kids next time".
The two royals got together to watch a rerun of Sunday's FIFA World Cup match (England vs Panama) at the Beit Al Urdan Palace in Amman. He was hosted by Crown Prince Hussein, 23, a member of the Hashemite dynasty Britain helped install in then-Transjordan nearly a century ago.