The Queen and the Duke of Edinburgh met the climber on the last day of their state visit to the islands.
They have faced some opposition from republicans and are expected to run into more controversy when the royal party moves on to Australia on Wednesday.
But Sir Edmund, 82, who met the Queen at a VIP garden party in Auckland, said: "Most people much prefer to have a Queen as head of state rather than a broken-down old prime minister.
"I would certainly like to see a continuation of the monarchy in my lifetime.
"The present situation is a very satisfactory one and I certainly support it."
It was on the eve of her coronation, on 1 June, 1953, that news reached London of Hillary's triumph.
Public support
With Sherpa Tensing, in a British-led expedition, he had made it to the top of the world's highest mountain.
Since then Sir Edmund, who lives in Auckland, has met the Queen on several occasions, including when he was knighted by her in 1953.
New Zealand's republican Prime Minister Helen Clark was also at the reception at Government House.
She has predicted that New Zealand will "inevitably" become a republic.
But a recent opinion poll found that 58% of people in the country wanted to keep the monarchy.
The Queen was greeted with crowds of several thousand cheering people on a walkabout in Auckland.
She left a barge after viewing one of the best-kept secrets in sport - the secret keel of Team New Zealand's 2003 America's Cup yacht.
The keel was shrouded from the media and public gaze by a giant curtain.
The couple will return from their 15-day tour of the three realms of Jamaica, New Zealand and Australia on 4 March.