Mr Middleton said he and his wife Alizee had settled into their new life as parents after their first child arrived a few weeks ago.
Posting on social media, Kate’s younger brother wrote: “He has been in our lives for just a few weeks but they have been the most special of my life getting to know our beautiful baby boy.
“No matter how prepared I thought I was… I was not prepared for the overwhelming emotion of meeting Inigo for the first time and the love for my darling Alizee as we became three.”
The newborn youngster is the Princess of Wales’ nephew and a cousin of Prince George, Princess Charlotte and Prince Louis.
Mr Middleton shared images of Inigo, wearing a knitted blue bonnet and cardigan, lying in his pram, near four of Mr Middleton’s dogs, and also being held by his father in a baby carrier.
Another photo showed Inigo’s tiny fingers touching a portrait of Mr Middleton’s beloved dog Ella, who died around nine months ago, and he paid tribute to her for helping him during his battle against clinical depression.
Mr Middleton has described how his dogs, and Ella in particular, helped him when he was facing mental health difficulties.
“We have settled into our new life as parents and I’ve told him all about Ella and that if it wasn’t for her he wouldn’t be here today and that we miss her greatly,” he said.
He said his remaining dogs had been “fantastic at welcoming their little brother into the pack”.
Mr Middleton also thanked the NHS and praised the team at Basingstoke hospital for their support with the arrival of Inigo.