The Duke and Duchess welcomed their first born on May 6, with the smitten new father introducing Archie to the world alongside proud mother Meghan when the newest addition to the royal family was just 2 days old.
Announcing his arrival on May 6, the duke said: "This little thing is absolutely to-die-for, so I'm just over the moon."
The Sussexes are raising seventh in line to the throne Archie at Frogmore Cottage - their new bolthole in Windsor Home Park in Berkshire, close to the Queen's residence Windsor Castle.
The Grade II listed 10-bedroom property is set close to the winding lakes, wooded mounds, glades, walks and bridges of the gardens at Frogmore.
Hands-on-father Harry has been relishing his new role as a parent after years of watching others bring up their families.
The duke said at Archie's first photocall at Windsor Castle: "It's great. Parenting is amazing.
"It's only been two-and-a-half days, three days, but we're just so thrilled to have our own little bundle of joy and be able to spend some precious times with him as he slowly, slowly starts to grow up."
The Duchess said of her husband and her son: "It's magic, it's pretty amazing. I have the two best guys in the world so I'm really happy."
Harry previously spoke about his desire to start a family, saying: "I've longed for kids since I was very, very young."
Father's Day in the UK - June 16 - is celebrated on the same day in the US - the third Sunday in June.
Harry's brother William marked Father's Day by sharing a cute picture with his son Louis and another with the Prince of Wales - but his other two children appear to be missing.
The Duke of Cambridge, 37, took to Instagram to share the adorable photograph of his youngest son Prince Louis playing on a swing rope, and another of his father, Prince Charles, 71, stepping off a helicopter, alongside the caption: 'Happy father's day.'
The Sussex's Father's Day post, in contrast, singled out only the Duke, rather than including the Prince of Wales, or the Duchess's estranged father Thomas Markle.
The photograph sent fans of the Sussexes into overdrive, winning a million "likes" in around three hours.
They will soon be offered another glimpse of baby Archie, it has emerged, with his christening scheduled for early July at St George's Chapel, Windsor Castle.
The Duke and Duchess will follow tradition by dressing Archie in a replica of the Honiton lace christening gown dating back to Queen Victoria, with a ceremony conducted by the Dean of Windsor and Archbishop of Canterbury, the Sunday Times reported.
The Queen is expected to miss the christening due to a diary clash, with the Prince of Wales, Duchess of Cornwall and Duke and Duchess of Cambridge due to attend.
Buckingham Palace declined to comment on the arrangements.
Clarence House also shared Father's Day social media posts, with one photograph of a young Prince Charles and Princess Anne with the Duke of Edinburgh, and a second of the Prince with his own two sons.
A message read: "To Dads everywhere, have a wonderful #FathersDay."
