This doctor's unique tradition certainly gives each baby he delivers a happy, exciting start in life. While his joyful singing might not be common practice (or as funny as this rendition or as unexpected as this one), the tune he sings is certainly a familiar one.
The immediately recognizable tune is not unique to the "Happy Birthday" song; it was part of "Good Morning to All," a song attributed to Patty and Mildred Hill. In 1893, Patty was a kindergarten principal in Kentucky and enlisted the help of her pianist and composer sister Mildred to come up with an easy song for children. Later, the tun was paired with the "Happy Birthday" lyrics; the first printed copy of the combination is from 1912, although it was likely common before then.