Harry, 36, who has been living in the US with his wife Meghan Markle, the Duchess of Sussex, 39, signed off the tribute using the words: "Per Mare, Per Terram".
Per Mare, Per Terram is the motto of the Royal Marines and is the Latin for "By Sea, By Land."
Philip had a great passion for the Navy and will be remembered for his service to it, as well as his devotion to his family and the country.
Some say he could have become the First Sea Lord had he not married Princess Elizabeth, who assumed the throne following the death of her father, King George VI.
At the onset of WWII, Philip started at the Royal Naval College at Dartmouth where he would go on to be named “best cadet” and eventually took command of a frigate, HMS Magpie.
With Britain at war in 1940, Philip joined the Royal Navy, serving as a midshipman on HMS Ramillies, and was posted to the Indian Ocean.
The Duke of Edinburgh was mentioned in despatches for his service during the war. He was a midshipman aboard HMS Valiant off the southern coast of Greece when he earned his honourable citation.
In his tribute, Harry also said his grandfather had a "seriously sharp wit" and could hold the attention of any room "due to his charm".
As well as being remembered as the longest-reigning consort to the Monarch, "he was my grandpa", Harry said.
"He has been a rock for Her Majesty the Queen with unparalleled devotion, by her side for 73 years of marriage, and while I could go on, I know that right now he would say to all of us, beer in hand, 'Oh do get on with it!'"
Harry added: "So, on that note, Grandpa, thank you for your service, your dedication to Granny, and for always being yourself.
"Meghan, Archie, and I (as well as your future great-granddaughter) will always hold a special place for you in our hearts. Per Mare, Per Terram".
Prince William, the Duke of Cambridge has also paid tribute to Philip. He and his brother are the first of Philip’s grandchildren to publicly honour his memory.
William spoke candidly about Philip’s relationship with Kate and expressed his gratitude for the “kindness he showed her”.
He also highlighted the special bond between the duke and Prince George, Princess Charlotte and Prince Louis, who would have “special memories” of being picked up in a horse-drawn carriage by their “great-grandpa”.
William pledged to uphold his grandfather’s wishes and continue, along with wife Kate, to support the Queen and “get on with the job”.
Over the weekend Philip’s four children spoke movingly about the loss of their father and how the Queen was coping after her husband of 73 years died peacefully on Friday.
Philip’s funeral will be held at St George’s Chapel, in the grounds of Windsor Castle, at 3pm on Saturday, 17 April and will be shown on television.
Additional reporting by Press Association