"[We're] absolutely thrilled and look forward to seeing him in the next few days when he's quieted down," William said of his nephew.
"I'm very pleased to welcome my own brother to the sleep deprivation society that is parenting," he jokingly added.
Markle gave birth to a son Monday after announcing her pregnancy in October. Harry told reporters he is "over the moon" about his baby boy's birth.
"Mother and baby are doing incredibly well," the new dad said. "It's been the most amazing experience I could ever possibly imagine."
"I'm so incredibly proud of my wife," he added. "As every father and parent would ever say, your baby's absolutely amazing. This little thing is absolutely to die for so I'm just over the moon."
William and Middleton themselves are parents to three children, 5-year-old Prince George, 4-year-old Princess Charlotte and 1-year-old Prince Louis. The couple shared new photos of Charlotte last week in honor of her fourth birthday.
William and Middleton had posted pictures of Louis ahead of his first birthday in April. Harry and Markle wished the prince a happy birthday on Instagram, adding, "Sending lots of love from both of us xo."