The 30-year-old McBusted singer, whose wife Giovanna Fletcher gave birth on Tuesday 16 February, gushed over his baby's tiny feet.
"Can't believe these feet are a week old already," he wrote. "Happy first week in the world Buddy."
Fletcher, who is also dad to two-year-old Buzz with wife Giovanna, has shared a few photos of Buddy in the past week, but none have shown his face.
Just a few days after Buddy was born, Fletcher posted a photo of his tiny hands wrapped round his finger. It seemed to show that the baby had a head of dark hair.
The dad-of-two has also shared photos of Buddy sound asleep in his crib and moses basket.
Giovanna Fletcher has teased fans in a similar way, sharing a selfie showing only the top of Buddy's head and full view of their cat.
The mum recently opened up about her breastfeeding journey, revealing she is going to be pumping and freezing her milk.
"I became quite obsessed with pumping and freezing my milk with Buzz," she wrote on Instagram.
"I loved looking in the freezer and seeing that it was well stocked 'just in case'. I had a really tricky breastfeeding journey with him, but so far so good with Buddy.
"I feel very lucky as I know it's not so easy for some and I really hate the whole bottle vs breast debate.
"Don't let others make you feel guilty about whatever choices you make. Happy mum, happy baby."