Logan is a defender capable of playing as a full-back or centre-back who made his full debut at under-21 level during a 4-1 win against Stoke City in February.
The 19-year-old, who hails from North Shields, started in all six of the Magpies' UEFA Youth League matches last season as well as making eight outings in the Premier League 2.
On signing his maiden pro deal, Watts said: "I was over the moon (to sign a first professional contract) because, ever since I joined the club at under-12 level, that's what I've been aiming for.
"Once I got a scholarship, I was pushing for that pro contract and then when I signed it, I was delighted."