In a statement released April 11, the 70-year-old said she and the family have been grateful to receive "messages and memories of so many people whose lives he also touched" and pointed out death is always difficult to deal with, no matter the circumstances.
"You know it's going to happen, but you are never really ready," she wrote. "My father has been my teacher, my supporter and my critic, but mostly it is his example of a life well lived and service freely given that I most wanted to emulate."
The Princess Royal, who was particularly close with her father, added that the Duke of Edinburgh had an "ability to treat every person as an individual in their own right with their own skills," and it was readily apparent in his work.
"My father has been my teacher, my supporter and my critic, but mostly it is his example of a life well lived and service freely given that I most wanted to emulate."
A message from The Princess Royal following the death of The Duke of Edinburgh: https://t.co/LqX46BeIVhpic.twitter.com/YkZxOTSrUM
— The Royal Family (@RoyalFamily) April 11, 2021
"I regard it as an honour and a privilege to have been asked to follow in his footsteps and it has been a pleasure to have kept him in touch with their activities," she wrote of his work and patronages. "I know how much he meant to them, in the UK, across the Commonwealth and in the wider world."
She ended her message by saying the whole family will miss her father, who "leaves a legacy which can inspire us all."
On April 10, Prince Charles, Philip and the Queen's eldest child and the heir presumptive, made a statement via video in which he called the duke is "dear Papa" and praised his dedication to Her Majesty, family, country and Commonwealth.
"As you can imagine, my family and I miss my father enormously," he said. "He was a much loved and appreciated figure and apart from anything else, he would be so deeply touched by the number of other people here, and elsewhere around the world and the Commonwealth, who also, I think, share our loss and sorrow.
"My dear Papa was a very special person who I think above all else would have been amazed by the reaction and the touching things that have been said about him and from that point of view we are, my family, deeply grateful for all that."
On April 11, Prince Edward, the Countess of Wessex, Lady Louise Windsor and Prince Andrew stepped out to a church service in Windsor in which they thanked the public and royal workers for their support during this difficult time.
Edward, who is the Queen and Philip's youngest child, said his father's death had been very difficult to process.
"It's been a bit of a shock," he said to reporters outside The Royal Chapel of All Saints. "However much one tries to prepare oneself for this, it's still a dreadful shock and we're still trying to come to terms with that."
Sophie, Edward's wife of 21 years, added that Philip passed peacefully.
"It was right for him," she said. "It was so gentle. It was just like somebody took him by the hand and off he went.
"Very, very peaceful and that's all you want for somebody, isn't it? So, I think it's so much easier for the person that goes than the people that are left behind."
Philip will be laid to rest with a ceremonial royal funeral at St George's Chapel, Windsor on April 17 at 3 p.m. local time. Only 30 people will attend the service due to COVID-19 restrictions, and the Duke of Edinburgh's final resting place will be in the Royal Vault below the church.