The footballer and his wife sparked speculation that they had looked to one of Gerrard's sporting heroes for name inspiration thanks to the unusual spelling of the baby boy's first name.
Gerrard announced the happy news to his fans on Instagram on Saturday to say that his son had been born earlier that day, with a photo reading "It's a boy!" alongside the baby's name and date of birth.
The caption read: "Lio George Gerrard was born this morning at 4.55am. Both are doing great. Our family is complete."
Former Liverpool and England captain Gerrard has previously spoken of his love for Barcelona player Messi.
In November 2016 when Barcelona beat Celtic in the Champions League, he said: "(Messi is) an incredible footballer.
"I think the best ever, the best there will ever be.
"He's just an absolute pleasure to watch."
Gerrard and Curran, who married in 2007, have three daughters - 13-year-old Lilly-Ella, 10-year-old Lexie, and Lourdes, five.