The England international posted a tweet on Wednesday in which he proudly welcomed Vivienne Jane Kane to the world, describing the newborn as a "beautiful addition to the family".
Kane also said that he was "so proud of Kate Goodland for having the most amazing water birth with no pain relief at all", a comment that landed the 25-year-old in hot water on social media.
Broadcaster Julia Hartley-Brewer led condemnation of the footballer's comment, writing: "Why on Earth should anyone be 'proud' of not having pain relief while giving birth? Utterly absurd! There's a reason why women get pain relief during labour: because it bloody hurts."
Kane has since responded online by posting: "Let's get one thing straight, any women can give birth however they would like. I'm very proud of my fiancee for doing it the way she wanted and we both learnt a lot from hypnobirthing."
Vivienne is the second daughter for Kane and his fiancee, with first-born Ivy entering the world in 2017.