The prince joked about his eye patch
The prince, sporting an eye bandage to cover a recent injury, joked that he would have to return to see the other half of treasures when it was healed.
The galleries, part-funded by £16m from the Heritage Lottery Fund, will open to the public on Thursday - the day the museum also introduces free admission for all visitors.
The new venue, consisting of 15 extra rooms, house one of the best collections of national treasures and mark the museum's most ambitious project in 50 years.
Wearing a large white bandage over his left eye, Prince Charles spent an hour touring more than 3,000 exhibits at the V&A on Tuesday.
They include some of the best examples of historic British furniture, textiles, costumes, ceramics, glass, jewellery, silver, paintings and sculpture.
'Talking to trees'
The prince told an audience of about 1,000 V&A supporters that he injured his eye while "talking to trees".
He said: "I must apologise for this
evening looking like someone who has been discharged prematurely from hospital.
"But during a long career of talking to trees you inevitably come across the odd argumentative arbutus or a cantankerous Quercus.
"In my present unjocular condition I regret to tell you that I only saw the Victoria half of the gallery. I will have to come back and see the Albert bit later."
'Ambitious project'
The galleries aim to tell the story of British culture from the days of Henry VIII to the reign of Queen Victoria. and mark the museum's most ambitious project in 50 years.
Highlights include Henry VIII's writing desk, James II's wedding suit, the Three Graces by Canova and the famous Great Bed of Ware, mentioned in Shakespeare's Twelfth Night.
Chief curator Christopher Wilk said the prince was particularly interested in the film footage of Queen Victoria, the 18th Century white and gold Norfolk House music room and a sculpture of a Newfoundland dog called Bashaw.
While most of the items in the galleries can only be viewed, the prince got his hands on a Victorian corset as part of a demonstration exhibit.
The Poet Laureate, Andrew Motion, wrote a poem, entitled Just Looking, to mark Tuesday's event.
Mr Wilk said the prince seemed to show no ill effects of the eye injury he sustained while sawing a tree in his garden at the weekend.
"He kept asking questions and looking at things and seemed delighted about what he saw," said Mr Wilk.
"He has a keen interest in the history of British craftsmanship and its relevance today."
The V&A's director, Mark Jones said: "We are delighted that the galleries will be free from the day they open so that everyone can enjoy these treasures."
Every major name in the history of British design is represented, including William Morris, Charles Rennie Mackintosh and Liberty's.
'Truly exciting'
Wherever appropriate, historic objects are compared with a modern-day equivalent to demonstrate how tastes have changed.
Culture Secretary Tessa Jowell said the galleries were "a truly exciting development".
She said: "I am particularly pleased that the museum has taken such pains to ensure that this collection can be understood and enjoyed by everyone.
"The opening also marks the introduction of free access for everyone a step ahead of the country's other national museums which follow shortly on 1 December."
The British Galleries were originally installed after World War II as the English Primary Galleries and have not been subject to major renovation since that time.
Their redevelopment is being seen as a flagship for future projects in other parts of the museum.