The Dutch royals were criticised for travelling during the coronavirus pandemic after Prime Minister Mark Rutte urged the nation to avoid unnecessary travel and issued a partial lockdown.
Speaking from their royal residence Huis de Bosch in The Hague, the king said on behalf of himself and his wife: "It hurts to have betrayed your trust in us."
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He added: "Even though the trip was in line with the regulations, it was very unwise not to take into account the impact of the new restrictions on our society.
"Our own decision to return was made with the realisation that we should not have gone."
King Willem-Alexander continued: "From the start of the coronavirus crisis, we have done our best to find space - within the limits of the coronavirus policy - and to be there as much as possible for everyone who is looking for support in uncertain times.
"It is a difficult time for everyone. A time of lack, limitations and worries. From fear, anger and insecurity too. We have heard the poignant stories in many encounters, in person and digitally. We feel connected with you and with all those people who have been directly or indirectly affected."
The royal couple spoke from their official residence
The message ended: "We will continue to work with you to get the coronavirus under control. So that everyone in our country can then resume normal life as soon as possible.
"That is the most important thing now and we will continue to do so, to the best of our ability. We are involved. But not infallible."
The Dutch royals released a statement on Friday night after arriving in Greece, which said: "We will abandon our vacation. We have seen people's reactions to media reports. And they are intense, and they have affected us."
King Willem-Alexander, Queen Maxima and their youngest daughter Princess Ariane, 13, are understood to have returned to the Netherlands on Saturday.
According to local media reports, the couple's eldest daughters, Princess Catharina-Amalia, 16, and Princess Alexia, 15, stayed in Greece for several more days as they could not get return plane tickets straight away.
The Dutch royal family pictured in July
Though the Dutch royals didn't break any of the Netherlands' lockdown rules, the government had discouraged unnecessary travel.
Officially, the region of Greece where the family's villa is located is under a mild Code Yellow travel warning from the Dutch Ministry of Foreign Affairs, meaning people from the Netherlands can travel, but the risk remains.
However, the coronavirus road map released by the government last week made it clear that considering the severity of the situation in the Netherlands, all unnecessary travel plans should be scrapped.
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